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0:01
[Music] okay today we'll be doing 24 rigorous limit proofs and of course we'll be doing them in one take and I have a file in the description for all the questions and make sure you're also know the four rigorous limit definitions in the previous video and after this video you will be much more comfortable for your University level Calculus class will real analysis class when you have to write a proof for limits so with all that being said let's get started with number one we
0:30
have the limit as X approaching 4 of 2x minus 3 is equal to 5 and this is how I like to remember
0:37
five years finite likewise this 4 is finite this is the typical Epsilon Delta definition later we'll be dealing with some Infinity here or Infinity here so make sure you know all the difference again check out the previous video with all the definitions that you should know before this video so I'm just going to write down the proofs firstly of course write down PF and for all this the four words will help you given choose support
1:03
check here will be given again because 5 is finite so we say give an Epsilon greater than zero and then second word is choose we'll be using Delta because 4 is finite choose Delta to be something that I don't know yet so I'm just going to leave it for now and then we are going to suppose
1:25
okay X is approaching 4 so we want the X is not too far away from four so the distance between X and 4 has to be small how small is small well less than Delta so we suppose the absolute value of x minus 4 is less than Delta and remember we don't want X to be 4 so we put this to be greater than zero suppose this happens then we hope to see that the distance between the function and the limit is less than Epsilon but we don't know
1:57
yet so we'll check it so after value this minus that we have the 2x minus 3 and then minus five
2:08
no hopefully this is less than absolute I'm going to show it usually we can do some algebra right here you see -3 minus five is minus eight and then we can factor out two absolute value of 2x minus eight if I draw the two we can put a 2 before the absolute value and then we still have that absolute value of x minus 4. all right this is okay but we maintain this absolute value and now check this out absolute value of x minus four is less than Delta go back to this and then
2:41
we can rewrite this as less than Delta thanks to our assumption so we have 2 times Delta now and keep in mind at the very end we want to end up with just Epsilon so by looking at this Delta should be equal to what well it must be 2 on the bottom so that it can cancel with this two and then the Epsilon on the top yeah so Delta should just be Epsilon over 2 and we can just come back here and put down Epsilon over 2. and again because Epsilon is positive that means our
3:16
Delta is positive yeah so this is legitimate and right here Delta is Epsilon over two so we write down equal and this is it once we're done make sure that you put box and then shade this in
3:31
this is less than Epsilon all right so that's the first one and we have 23 more to go so here we go
3:44
all right let's get into some study series business number two I have the limit as X approaching five and on the bottom I have x minus 3 on the top of one and this is one half it's a fraction case right so what do we do well again finite finite so this is again a Epsilon Delta definition different Epsilon greater than zero choose Delta we don't know yet just leave it and then we are going to suppose the distance between X and 5 is less than
4:27
Delta and make sure that is greater than zero and then we are going to check this minus that in the absolute value is less than Epsilon here let's do some algebra and the way I like to do it is let's just multiply the denominators so we have 2 times x minus three and then I do this times double choose two and I do this times that so I have minus X and a minus minus 3 which is plus three so that's what we have still maintain the absolute value
5:04
now on the top though we see that we have 2 minus X plus three so that's minus X and then plus 5. we can factor out a negative and the negative in the absolute value doesn't matter so in fact let's write this down we have one half it's just a constant so I'll keep it like this and then again this right here I can write it as absolute value of x minus five after factoring your negative by the negative doesn't matter in the absolute value and then we have the x minus 5.
5:35
and then we have the x minus 3 on the bottom still in the absolute value so I'll just put down times when one over absolute value x minus 3. okay so yeah as I told you once we have the reciprocal we're getting to study more serious business so what do we do though here notice that we have absolute value of x minus five we can come back to our assumption here we know that's less than Delta so that's very nice so I'll just put down this is less than Delta
6:07
and then we have a one half at the front and now the question is what do we do with this though one over the absolute value of x minus 3. it looks so we are stuck here's the thing though usually when we go from here to here hopefully we can just have an expression in terms of Delta and then later on we'll be able to figure out what Epsilon should be we will be able to figure out the attention B in terms of Epsilon by the way we need to have a Bound for this here's how we do it
6:38
we want to have another condition on Delta so that we can work with this inequality usually let's just come back here right away and then put down minimum and you can pick some numbers that you want easy enough number just one usually works out really well so I'm just going to pick one sometimes one works perfectly but not all the time there will be example that we cannot use one but that will be for later on for now let's put down one with that being said we know that
7:11
we can just look at this and say absolute value of x minus 5 is less than one open this absolute value we know x minus 5 is in between of negative one and one and then we'll need to somehow get to this so right here let's just go ahead and add two to everybody because we want to end up with x minus 3 somehow so this is one and then we have x minus 3 and this is 3.
7:41
now we have this on the bottom and also the absolute value so what should we do first though
7:47
I'll tell you let's take the reciprocal first so if we take the reciprocal then we are going to get notice they're all positive the reciprocal of this is one and the reciprocal is just one over x minus 3 and the reciprocal of this is one of x minus 3 and this is one over three because the O positive we can just flip the inequalities once we have this take the absolute value so I can apply the absolute over here absolute value of one doesn't matter so we are just looking at
8:19
one over perhaps the value of x minus 3. we just picked a bigger number therefore has the bigger absolute value technically one we say this is less than one and then that's the bound that we want
8:34
let me just show you though what if you took the absolute for you first imagine right here if you took the absolute value then you are saying that absolute value of x minus 3. again you pick the bigger number that has the bigger absolute value which is three so you're saying that this is less than three which is true and then if you take the reciprocal here you are looking at one over absolute value of x minus three and you compare that with one over three
9:03
because they are both positive and you'll just have to flip the inequality but what's wrong with this though well remember we want to show that this thing is less than blah blah blah and then Epsilon at the end we want to have less than if you have a greater than here if you put it here that's like they're fighting less than and greater than they are fighting no don't do that it's like you're fighting with little brother or a little
9:30
sister no don't do that so don't do this right this is not the way to go about it but this is so keep in mind take the reciprocal first and then the absolute value then we can come here and say this thing is less than one we have a less than right here already so I can just say put on a one like this all right now once we have an expression in terms of Delta only then we want to figure out what Delta should be in terms of Epsilon so that we can end up with Epsilon
10:06
one half times four will be Epsilon well we know one half times two Epsilon will be Epsilon it can store very nicely so we want the Delta to be two Epsilon right we want to have this condition as well so we come here and then we put down two Epsilon
10:31
so once you have the one you get to use this right here and the ones will have the two Epsilon you get to say this right here here's a small thing now before we
10:42
um finish because we still have this thing right we still have that little part that we have in viewing what should we put on right here we'll have a look not equal sign because right here we are saying Delta is the minimum of one word to Epsilon this or that but either way we know this is going to be less than or equal to 2 Epsilon
11:07
all right because it's the minimum of this for that so either way it's going to be either equal to 2 Epsilon or less than two Epsilon so Delta is less than or equal to 2 Epsilon right here we put down less than or equal to with all that we are done put a box right here and shade it in and let me tell you the moment you put on the box and shading it this is much better feeling than you put on a plus c after you finish your integral trust me right so number two okay continue
11:44
I don't know if you guys will disagree of what I said or not but
11:49
okay to be honest they both feel great right plus see what the box they both feel great but I will give a
11:59
I will choose the box over the place here all day because it's just it's just that anyway number three here the limit and I will still give you guys the same function still one over x minus three but this time though we are going to take the limit as X approaching Infinity of course if you work this out it's just one of infinity the answer should be zero how do we prove it here we go p f the limit is zero which is finite so we use Epsilon we want
12:33
to make sure that the distance between this and that is less than Epsilon right so given
12:39
Epsilon greater than zero choose
12:44
well X is approaching Infinity so don't do Delta we are going to use capital n but what's capital angle don't worry we don't know yet just write it down for now we'll figure out later
12:59
when X is approaching Infinity we are going to write down suppose but don't say the absolute value of x minus infinity that doesn't make sense X approaching Infinity means that X is going far far to the right it is big enough how big X is greater than diabetic if x is greater than n then we hope to show that the distance between this and that is less than Epsilon absolute value and I will still write down our function and then
13:34
our purpose to write down the minus zero but I don't we know minus zero doesn't matter
13:40
so this is really just x minus 3 on the button and then we have a 1 on the top and now what though [Music] well we are going to come back to our condition we have X greater than n so we have to somehow use this here's the thing when X is greater than n and X is approaching Infinity
14:00
right here let me just put on a small condition this is greater than three I want to have that but cannot really say so yes this thing right here is going to be an expression sample be greater than n b greater than three just trust me but the reason I need you to say the above greater than 3 is because once the above greater than three then we can say x minus 3 is greater than n minus 3 and they are both positive and then we can take the reciprocal on both sides
14:32
and we get 1 over x minus three one over n minus three they are both positive so we can just flip the inequality like so and that's exactly the correcting quality that I want
14:45
you don't really need to do this all the time if you know that X is greater than n is greater than 3 you can do the following you know this is passed here you can just replace the X with n so you write down 1 over n minus 3. can we make a computer set yes because X is bigger than n that means x minus 3 is bigger than M minus 3. if the denominator is bigger than the other one and they have the same numerator then the whole fraction here is going to be smaller than this
15:24
fraction so that's okay right so you don't really have to write this down but of course it helps now we're having expressing tens of n just the earlier you want to have an expression in terms of Delta only so you can figure out what Delta should be in terms of Epsilon right here we can figure what entropy in terms of Epsilon to do so you just write it down on the side one over n minus 3 you put it to B Epsilon multiply this divided by that we know one over Epsilon
15:56
equals n minus 3 as 3 on both sides capital N equals 1 over Epsilon greater than plus three
16:05
anyway though you see this is the under one we want one over Epsilon plus three I told you I told you so just note that put that down for now and then hope for the best so that being said I'm just going to put this down for n so we have 1 over 1 over Epsilon plus three and then minus three because n is equal to that here it will be an equal sign work this out of course we get Epsilon
16:41
done right done so this is how you prove a limit when X goes to Infinity
16:51
all right so let me know if you guys are really doing absent data definition or Epsilon and definitions all that in your Calculus class usually at a university they will teach this or if you are taking a real analysis class which is upper level
17:09
um nav class then you do this or like a on a Calculus class all right number four
17:18
here is a limit and again I'll still give you guys the same function 1 over x minus three and another interesting number that we look at is if x is approaching three Plus if x is approaching 3 plus this thing will approach Infinity right no just kidding I put down Infinity I want to say infinity
17:46
all right here we go the limit is infinity so what do we do we use Big M right given
17:59
Big M we want because we want to show that the function is you know going up and up and up bigger than any M that you can possibly set so given M greater than zero but X is approaching 3 plus this is finite so it's Delta we will say choose
18:19
Delta to be something we don't know yet just leave it then we suppose
18:26
and now here's the deal if we have X is approaching 3 this means we are looking at three in the middle and we can go from the left we'll go from the right Delta distance away so we can get 3 minus Delta here or 3 plus Delta here and X is somewhere in the middle
18:49
but that's when we have just a 3. here we have a plus if we have X is approaching 3 plus this means we're just going towards three from the right hand side so here's three coming from three from the right hand side which was care about to have three plus Delta on the other side so here I'm going to write down the distance between X and three so I'll write down x minus 3.
19:18
is less than Delta and again we don't want X to be exactly equal to three so put that down you see the difference is that if you have X approaching three plus you don't want the absolute value or you have the absolute value you go both directions and of course similarly when you have X approaching 3 minus that means 3 right here and you go to the left only right so this is three minus Delta like this then in that case your right x minus three is in between
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of zero and negative Delta just from here to the left yeah so that's the small difference
20:00
that being said again the key is play around with the inequality make sure that this time though we want to get our function so let's check does happen our function is 1 over x minus three we're going to be going up and up and up
20:20
and you don't need absolute value going up is passing so you want to have this we want to show that this right here is bigger than our capital and now we said earlier but I don't know you I cannot put that down so let me race
20:37
so what do we do well don't worry sometimes this kind of limit can actually be easier here x minus three a sympathy of zero and Delta this is positive so can we just do reciprocal yes of course just focus on this part right don't do specific cover that if we take a look at 1 over x minus 3 compared with 1 over Delta well flip the inequality we want the greater than this time we want to Greater that and we do have the greater than
21:11
so we can right away say it's greater than 1 over Delta right away very nice and now when you have an expression in terms of just Delta you can just go ahead and set whatever that is to be whatever we put down earlier which is capital M so for Delta in terms of n Delta is just 1 over m
21:37
put it here one over Big M and then put it here to finish so we have this equal 1 over 1 over Big M of course this right here is just Big M and we are done Papa Papa so so far number two is the hardest one right but if you practice of all others seriously after the 24th rigorous then proves you will be able to write proofs as well and your University Calculus class for real analysis class will be much better much more fun all that good stuff okay number five
22:22
here we have the limit this time we have X approaching infinity and we have 3x over
22:30
2x plus 1. this right here x to the first power x to the first power so just three over two nice this is the question that you will love so much in your Calculus one class if just the computation but to talk about the proof well that's the p f this is finite so use Epsilon yes
22:53
given Epsilon greater than zero this is approaching Infinity so we use big n yes choose
23:07
begin to be something well here we are going to suppose
23:14
when X is bigger than big n then we hope to show that this minus star in the absolute value is less than Epsilon so absolute value we have 2X plus 1 3x and a minus this
23:32
all right do the algebra so I'm just going to multiply this and that on the bottom so 2 times 2x plus 1 and I will do this time step which is 6X and I'll do this time step which is negative 6X and then negative three I still need to put down the absolute value but we notice this then that cancel absolute value of negative history doesn't matter it's just read and then we have two and then 2x Plus 1. now here's the thing do we need the absolute value
24:09
no because if x is approaching Infinity then you hope that X is big enough right so let's just put down this right here it's greater than zero but technically as long as it's greater than negative one half it's okay but I'll still just put on great density rule because if this is greater than zero of course it's greater than negative one half with that being said we don't need absolute value
24:32
and this is equal to that now how can we proceed here's the deal we know that X is bigger than n
24:45
if you replace the X with n then you can get the following you get 3 over 2 times 2 and and then plus one and that's great because we can actually make a connection between this and that this has to be less than that because the denominator here is bigger than this denominator due to the fact that X is greater than our assumption here and then the three is times the same once you write this down of course when we have an expression in terms of n already you can just
25:22
do some scratch work on the side real quick put that down and then make it equal to Epsilon and then so for n in terms of Epsilon but this is a little bit too much work or the solving point I'll show you guys something inside the better keep in mind if you can write down each and every inequalities legitimately then the proof is also legitimate I have done a video going over the details on that so check that out if you need to so here we go how am I going to do it
25:59
denominator the smaller the whole fraction is here we have 2 times 2x Plus 1. Let's ignore the plus one compare this at 3 over 2 times 2x [Music] can we say something about this and that yes we can say this is this then that yeah sure thing so as you can see we can actually just say this right here is let me just put down Less Than 3 over 2 times 2x which is 4X like so
26:36
slightly easier but you will see that this technique will actually be used for eight round two so just kind of get used to that and now we know X is greater than n so again I'm just going to replace the forwards n and then we will get 4N right here I would just say this is less than oh sorry I shall do this for you guys three on the top four here replace the X with n and we can say this right here is less than that
27:05
now with this right here it's just in terms of n three over four n equals Epsilon and then solve it real quick if you want you can do the following two three over four is less than one so it's actually okay to say this right here is less than one and then over n so why not
27:31
so the easiest and to get is just one over n you make it equal to Epsilon so n equals 1 over Delta 1 over Epsilon we put n to be one over Epsilon Epsilon is greater than zero so n is greater than zero so this is true and then right here we can just say this is equal to 1 over 1 over Epsilon which is just Epsilon and then with that we are done this is less than Epsilon right so just like that
28:18
all right so we have cover pretty much all the situations but there's one thing one situation we haven't covered X approaching infinity and limits Infinity but that's coming soon let's do another harder one I will say number six here's a quadratic case but that's when X approaching two and we have 3x squared work that out you get 12. so finite finite this is the classic Epsilon Delta definition so here we go start off by saying given
28:57
Epsilon greater than zero here we are going to choose the outer to be something this is X approaching 2 without the plus or minus so we say suppose
29:10
the distance between X and 2 and you keep it in the absolute value is less than Delta and then right here make sure you also have that we are going to check
29:22
the function is close to the limit in the sense that absolute value of our function is 3x squared minus our limit hopefully this is less than Epsilon all right algebra time we can factor out a 3 and x square minus 4. x squared minus 4 can be fantasy again and here's the thing the absolute value of a product is the product of the absolute values so let me just say absolute value of strip which is just 3 and then absolute value of the first factor which is x minus 2 and
29:58
then the second Factor like that so from here to here is usually pretty cool you can just walk down real quick all right we want to change all the terms and somehow give the bound right now this we can use that right away this is less than Delta so we can say that's less than
30:19
3 is just V and this right here is Delta pretty good now we have a small trouble what do we do is this well let's give Delta sound number so we can work with this inequality more right same style put down minimal and let's use one again
30:42
with that being said we can look at this part and say absolute value of x minus 2 is less than one open that we get x minus 2 is in between of negative one and one and we want to get to plus 2 right so let's just go ahead and add four um not both sides but like to everybody so this is three and then we have X plus 2 here and then this is 5. now apply the absolute value once we have this we know absolute value of x plus two you pick the bigger number
31:17
that has the bigger you pick the number that has the bigger absolute value so that's step five
31:23
so once we have that we can say this thing is less than five perfect
31:30
that being said three times five is fifteen and to end up with Epsilon or at the very end we must make sure that Delta is 1 over 15 times Epsilon so put on Epsilon on top so I will come here Epsilon over 15. close this and we know this is going to be less than or equal to Epsilon over 15. so this is less than or equal to and then we have the three we also have the five and we see that they cancel out with doubles Epsilon perfect just like that [Music]
32:22
all right number seven
32:26
here we have the limit as X approaching Infinity oh oh infinity and we have a function in the square root square root of 3x plus 1 work that out you get infinity because square root function in this case is an increasing function to Infinity
32:45
good this is the first time that we are dealing with both Infinity so use Big M because of this infinity and we also use big n yeah first we say different Big M greater than zero secondly choose begin to be something we don't know yet then suppose
33:08
X is far away right meaning it's bigger than our weekend big enough right now check our function it's also big enough in a sense that our function is bigger than M but how do we do it
33:28
well here is the thing it's really fun check this out again work with inequality we have a plus one here in a square root and we now square root the increasing function so if we just take away the one I can just say this right here compare with square root of 3x this is bigger than that that's true so you can ignore the plus one in fact if you really want to because again this is increasing function and X is greater than n you could have say square root of three X plus one
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is bigger than square root of three n plus one and then once you have this set equal to M and then solve for n in terms of M that's fine but this is easier and if you really want to square root of 3 is bigger than 1 you can know the 3 as well should we do it sure why not this is bigger than square root of x because square root of 3 is bigger than one and sometimes you can say not just denote that square root of 3 is bigger than one this is one point six seven something so this
34:43
important is still true and you can also say because square root is the increasing function
34:52
square root of x is increasing so take away the plus one this is true yeah so just like that now again because curve X is increasing I can just replace the X with n so we have that and then this is still greater than looking at square root of N I can say square root of n set e to b m
35:19
solve it and equals m Square that's the end that we want M squared this is equal we have the square root we have the M square and this is just simplified to n done
35:38
okay so this is legitimate you may see some people they have different and as long as all the inequality works all the proofs like it's legitimate cool [Music]
35:58
let me check if this is still recorded perfect number eight
36:06
I'm going to give you just the same function again still limit square root of 3x plus 1 but this time let's take the limit as X approaching 5. work that out as fifteen plus one sixteen take the square root get 4. all right finite the finite so Epsilon Delta definition
36:26
given Epsilon greater than zero and then we choose
36:32
Delta to be something we don't know yet leave it suppose
36:38
absolute value x minus five is in between of zero and Delta and we are going to check the distance between our function square root of 3x plus 1 minus four right this minus that hopefully this can be shown to be less than Epsilon
37:02
now we're kind of stuck because how can we make a connection between this and that well don't worry keep in mind our goal that is back in the regular calculus so you have to compute limits involting square roots such like this the limit as X approaching now say five square root of x plus four minus three and then x minus five something like this what do we do we multiply the type and Bottom by the conjugate right right here let's do the same thing
37:36
so this is equal to I will keep this as healthy as square root of 3x plus 1 minus 4. but on the top I will multiply with square root of 3x plus 1 plus 4 and now of course we also divide that which is the square root of 3x plus 1 plus 4. yay work this out multiply this and that square root cancel so we just have 3x minus one and a minus this is just a minus B times a plus b so it's minus this thing squared which is
38:17
16. and if you look at the top this is just 3x oh sorry it should be plus one this is 3x minus 15 we can factor out three so this is equal to we have a 3 right here all the way the at the front and then we will have x minus 5. keep it in the absolute value
38:42
right and then lastly we still have this on the bottom but notice that this is always past the T1 actually now negative well yeah non-negative plus four is always positive so we have this over that square root of 3x plus 1 plus 4. so that is what we have
39:04
pretty good
39:07
and how can we continue we do have the x minus find the absolute value so me too can use this right away but you know what if you do that verb and then the bottom is doing to be biting us right so can we get rid of the button yes and this is how if you just ignore the bottom computer this with the top part which is three absolute value of x minus five can we make a comparison
39:35
yes and in fact this is less than that why
39:43
check this out we have this thing plus four the bottom this is now negative plus four for sure is bigger than one don't say bigger than zero this is true because we can say square root of three X plus one plus four is bigger than one right one is the key number here don't say bigger than zero here is the it's simple so suppose we have two numbers that say two and two but right here if I divide by a number bigger than one asset divided by six compare them which one's
40:21
bigger this one's bigger so I can still maintain that kind of inequality this is bigger yeah so that's what we want but if we have two let's say this is one and two let's say we have two and two
40:37
if I say that's divide this by a number bigger than zero why if I pick this is 0.5 what's 0.1
40:49
this is a positive number if I divided by that which one is actually bigger this is actually bigger now and that's not the inequality that we want all right so once you set this is greater than one you can actually know the bottom so we can just focus on the top remember the less the expression that you have the easier the proof will be so just kind of get rid of stuff once in a while all right with all that being said of course here comes to the best part we have three times
41:20
absolute value of x minus five we know that's less than Delta so this is three times Delta
41:30
at the very end we want to enter with Epsilon so Delta should be equal to what
41:35
dot Epsilon was three so I would just say this is equal to the three is still here but Delta will be Epsilon over three cancel cancel done don't worry I haven't forgotten to put Delta to be Epsilon over three now it's done just like that just like that and you might be wondering what if this is minus but let me tell you now actually think about it will this happen if this is a minus when we're doing these kind of things can we get minus on the bottom think about it the answer is
42:13
no all right because this is always a subtraction so the conjugate is always a plus so not possible
42:22
all right number eight
42:31
at number nine let's go back to our linear case but this time though let's say x is approaching a negative number X approaching negative 2 and then we have negative 3x Plus 5.
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work that out you get 11. this is a Epsilon Delta definition right so given Epsilon greater than zero we are going to choose Delta to be something and then suppose
43:04
the distance between X and minus 2 so we'll write absolute value x minus negative 2 so become plus two like this and then let's say this is in between of zero and Delta and we hope to show that the distance between our function which is negative three X plus five and minus 11 like this
43:31
all right can we do Algebra yes plus 5 minus 11 is just minus six and we can factor out a minus 3. so you can say minus three and then x minus
43:47
so it's minus six but we factor of minus three so this right here should be X plus two yeah and then I'll still maintain the absolute value for you guys but absolute value of negative three is just a 3 so we just need to look here three times absolute value of x plus two
44:05
absolute value of x plus two hey we can use that that's less than Delta so Delta should be again Epsilon over three just like the Delta W 30 but it's just coincidence so this is three times Epsilon offers 3 cancel oops don't don't say this is equal to a box you want to say this is equal to
44:42
[Music] let me tell you I have never done this many Epsom Delta proofs in my life what rigorous damage proof yeah today it's actually the second time but it's the same questions but anyway number
44:58
10. here let's talk about this one the limit as X approaching 3 and we have X over x minus two work dot o because three over one so it's just three okay this is
45:17
finite Finance so Epsilon Delta definition again in my opinion Epsom Delta definition could be the hardest one but of course some times it depends on the function this is a little bit tricky but have a look proof given Epsilon greater than zero choose Delta to be something I don't know yet suppose the distance between X and 3 is small enough meaning smaller than Delta but also make sure it's greater than zero check our function and the absolute value
45:58
minus our limit hopefully this right here is less than Epsilon so let's just go ahead and just work this out so we can just say we have the same denominator just x minus 2 and then this is X and I will do x times negative 3 which is negative 3x L to negative 2 times negative 2 which is Plus 6. still maintain the absolute value x minus 3x is just negative 2X and then we still have the plus
46:29
six good I can factor out a negative two absolute value of negative 2 is just two so let's just work that out right now we have a 2 at the front and with that being done we will have x minus 3 left inside right factor and then you just keep the absolute value so absolute value of x minus 3. all right so hopefully that's clear and all that stuff and then lastly we have the x minus 2 on the bottom so just say multiply by 1 over absolute value of x minus 2. [Music]
47:05
this is less than Delta so we can say that's less than 2 is still two Delta is here but what do we do with this yeah we have done this twice already go ahead come here and say Delta is equal to the minimum and then right here pick a number that you want already I told you that's just one so let's see why if we still use one so with that one right here we can go here and legitimately say absolute value of x minus 3 is less than one open this x minus 3 is in between
47:41
of negative one and one but we want to get to x minus 2. meaning we will just add one to all this so we are going to get 0 and this is x minus 2 and this is two and equality stays now what do we do
48:01
remember I told you guys to do what reciprocal on both sides first right and then take the absolute value
48:13
just imagine if I take the reciprocal then I get a 1 over infinity
48:25
I mean one over zero
48:28
can I do this one over zero and then this is one over two and then one over x minus two
48:35
like this well it's arguably true but like I need to find actual number to bound it so maybe I shouldn't take the reciprocal first if I take the absolute value first then what will happen
48:52
if I take the absolute value then I'm looking at absolute value of x minus 2 is less than two great and then if I take the reciprocal from here one over absolute value of x minus 2 great one over two great but this inequality becomes greater than n we don't want that we want to have just less than so this is not what we want and unfortunately this does not give us any good number so it's kind of like we're stuck
49:22
think about why this time the one doesn't work got it okay I'll tell you the reason that the one doesn't work is that remember once it's kind of like the distance from three away right
49:37
X is approaching three so if you do catch three if you go to the right if you go to the left one unit then you're on your left one unit what do we get we get two
49:49
what's x minus two zero doesn't work so in that case you see that the difference between three and two is one so in this case one it's a bad number to use don't use one uh so what do we do well don't worry about it same strategy but just use a different number let's say one half one third one over four one over ten up to you that will stay with one over two same idea once we put down the one number two here we can come here and say that's one over two
50:21
open the absolute value x minus three it's in between of negative one over two and positive one over two to get to x minus 2 I add 1 to everybody here then negative 1 over 2 is
50:38
going to be added with one so that's one over two x minus 2 in the middle and that's three over two
50:46
perfect and then we do the reciprocal then it's perfect now reciprocal of this is two reciprocal this is one over x minus two reciprocal of this is 2 over 3. flipping equality like so it's legit then I take the absolute value 1 over ABS to five x minus two is less than two
51:12
yay so I come here and say this thing is less than 2.
51:19
not being done two times two is four so we know that to enter with Epsilon at the end Delta better be Epsilon over four come here Epsilon over four
51:34
close this and we know to put out less than or equal to and let me just write down the two and the two and then cancel them out so this is how write the proof so that when people read from the beginning to the answer wow how did you know all this no I didn't know that ahead of time we have to work it out and just leave this for now a little bit and try to figure out what this should be in order to break down the good proof so very nice question of course I came up with this anyway
52:19
all right number 11 let's prove the limit as X approaching Infinity 1 over x square that's equal to zero it's true is finite it's okay let's just go ahead and start with given
52:39
Epsilon greater than zero here x is approaching Infinity so we say choose
52:46
big n and then suppose
52:52
X is greater than n hopefully we can show that check that's the value of our function
53:00
minus the limit is less than Epsilon well this time we can just take out the absolute value because -0 doesn't matter and then x squared is no this thing is always positive
53:13
cannot be equal to zero so just always positive like this hmm come into anything yes here's the thing X is greater than n we can say that's greater than zero you will see later it will indeed happen once we have this you can just work this out real quick x square compare to n Square
53:36
they're both positive so it's greater than take the reciprocal 1 over x squared 1 over N squared
53:44
this is the inequality less than so we can say 1 over x squared we can say that's less than 1 over N squared right or you can just argued in the sense that we know X is bigger than n x squared is bigger than N squared thus this whole fraction is smaller than this whole fraction well if you want to work it out like this that's fine too
54:10
now once we have an expression in terms of just n just put it on the side set it to be Epsilon and solve it so this means we have n square equals 1 over absolute
54:25
one over Epsilon and then take the square roots on both sides n equals 1 over square root of point just one so just that just a positive version of course come here 1 over square root of Epsilon of course it's greater than zero of course this is true so that's equal to 1 over 1 over square root of Epsilon and you square that and at the end you end up with Epsilon
55:14
[Music] right now number 12. here that's proof the limit as X approaching Infinity
55:24
square root x minus 3 this right here will be approaching Infinity as well so how can we make this work it's kind of similar to the square root it's important with the earlier but you'll see a study different here p f Infinity use M this Infinity is n so given
55:45
Big M greater than zero choose begin don't know yet suppose
55:55
X is greater than big n check our function which is square root of x minus 3 is greater than M don't need absolute value so now what do we do can we ignore the numbering side no this is -3 so I cannot say compare this with square root of x and then say that's bigger doesn't work but don't worry because this function is increasing function so if x is greater than n we can really just replace the X with n and then the minus 3 take the square root we know this is greater than that
56:38
this gradual function is increasing right so this is the increasing function
56:48
so you can do more calculus in that regard but I think this is pretty clear so once you have this right here in this case cannot really take away any numbers so we're just going to set this square root of n minus 3 into B Big M Square both sides and then plus 3 on both sides so n equals m squared plus three and that's all exactly what we need M square plus 3. right so right here we can just replace M square plus three minus 3 and then square root and this
57:27
is equal to work that out you get became done right I think this is pretty clear Sometimes some people may want you to you know write it down as this is greater than 3 and then just work out okay x minus 3 is greater than M minus 3 greater than zero and then you take the square root on both sides like this so square root of x minus 3 greater than square root of n minus 3. so yeah so it can't depends this might be even more career but that was pretty good too all right
58:11
we're halfway done only halfway done okay so it's probably going to take about two hours number 13.
58:25
I'll give you guys a pretty simple looking function X approaching 2 and then let's talk about X Plus third power for the first time plucking you can eat finally finite Epsilon Delta given Epsilon greater than zero we are going to choose the outer don't know yet then we are going to suppose
58:52
the distance between X and two so absolute value of x minus 2 is in between of zero and Delta and then we are going to check
59:05
this minus dying the absolute value somehow it's going to be less than Epsilon yeah okay here we go what can we do Factor it of course so Factor this as the difference of two cubes we get x minus two as the first factor and then let's just apply the absolute value for that and then for the second Factor we get x squared plus 2X Plus 4. this is good that's less than Delta done but this thing we have to figure it out a little bit we can do the same thing
59:42
that we have been doing before right whenever we have like a extra part I have like X and I would cannot really get rid of it right away in that case you should come here and write down Delta to be the minimum of something just use one because you know just no worry at all for this not being said we can say this right here is going to give us absolute value of x minus 2 is less than one open it we get x minus 2 in between of negative one and one and uh
1:00:16
you can think about how you can go from here to here right but I want to show you guys a little bit more I'm just going to bound the x value I will just add 2 to everybody and two to everybody
1:00:30
that will give me X in the middle and then this is one and that's three with that being said we can set up to 5x is less than 3. you can still try to square everybody and then add the 2X and then plus 4 something like that but sometimes you have like a weirder fraction equator function then maybe you can also try this but anyway though here this is how I want to do it for this time
1:01:01
earlier I used this yeah but as we can see here we have three terms and they are all adding so I want to take the advantage of the so-called triangle inequality meaning
1:01:15
we can break this down as the absolute value of the first
1:01:21
right it's all adding then you can just say this thing is less than or equal to
1:01:28
actually let me let me just let me just write down one more line to make it more clear absolute value of x square plus 2X plus 4 right and then this time this is less than I still have the Delta here but I will break it down by the triangle inequality absolute value of the first
1:01:50
but up to the value of x squared it's the same as ABS to five of X and then Square so I can put that down like this then we add it with absolute value of this the two can be taken out right here and lastly absolute for your fourth which is just four power steel QBs absolute value like so and this is the triangle inequality so I'll just put this down right here for you guys because of triangle
1:02:20
in equality which says the following for all real numbers m b when we have the absolute value of a plus b this is less than what equal to absolute value of a plus absolute value of B so technically I should have put down less than or equal to
1:02:41
and that's not equal to yeah sorry about that [Music] so it works wonder why because we know ABS to Phi of X is less than three so right here I can replace this as less than the altar is still at the front chilling but when we see up to the value of x we can replace that with three so this is three square and then plus two times this right here is three and then this is just four
1:03:11
this thing is equal to Delta times this is nine plus six is fifteen Plus four is nineteen all right so thanks to that this right here is just a number as well so I think slightly easier unless you really want to square both sides right here and then you add it just yeah
1:03:37
anyway times 19 and you can just set this to be Epsilon figures right but usually this step is just pretty straightforward this is just going to be put Delta to be Epsilon over 19. right this is
1:03:58
cancel cancel I will come here and then put the oops sorry I shouldn't say equal
1:04:05
I will say epsilon over 19. and and that should have been less than or equal to Delta is less than or equal to Epsilon over 19 whenever you stand minimum so just like that
1:04:36
right number 14.
1:04:42
limit as X approaching 6 1 over 2X and we are going to add one work that out you will get four here we go proof Epsilon Delta definition given
1:04:57
Epsilon greatness zero choose the outer to be something we don't know yet suppose
1:05:07
our distance the distance between x and six is in between of zero and Delta then we hope to show that the distance between our function which is one half X plus one and we're subtracting four like this is less than Epsilon all right do the algebra this entire just minus one so this is absolute value of one half X sorry 1 minus four is minus three
1:05:42
all right this and that they don't look the same here but check this out the coefficient right here is one half we can factor that out and when we do so we'll just have one half at the front and it absolute value and just think about it as three divided by one half so we actually have x minus six so it still works so we can say this is less than Delta
1:06:06
right and the figure of what Delta should be and I will just tell you Delta should be
1:06:13
two Epsilon because they cancel so I'll just say two Epsilon finally right on the Epsilon
1:06:24
just like that all right
1:06:30
it's actually quite a lot of questions a lot of writing but this
1:06:38
all right number 15. here let's take a look at the limit as X approaching Infinity
1:06:48
and on the top we have X on the bottom we have x squared minus one work that out you get zero zero is finite so we use Epsilon so p f given Epsilon greater than zero X is approaching Infinity so we want to choose a big n to be something then we are going to suppose X is big enough meaning bigger than the big end we hope to show the distance between our function and the limit zero
1:07:25
is less than Epsilon minus zero doesn't matter check this out though we will just have X over x square minus 1. how can we provide a condition here so we don't need to put on the absolute value
1:07:41
on the top add shop equivalent zero but here we have a minus one so we really want to have make sure X is greater than the square root of 1 which is just one put that down for now so we can just ignore the absolute value now what
1:08:02
can we Factor it this is x squared minus 1 so can we Factor at x minus One X plus one
1:08:10
maybe you can try it but no cannot do it all right I'm not going to do that and then let's see
1:08:18
can we somehow get rid of the minus one and compare this with just X over x squared yeah we can compare but this would be that being bigger than this though right so ah that's not the correct inequality that we want because when we have the Epsilon greater than zero we want to show this is less than Epsilon so that's the bad inequality so what we do
1:08:50
well here's the thing wouldn't it be nice if we can just somehow cancel the
1:08:58
X on the top let's think about it or maybe you want to try right here and then replace all the excess with answers so that's the n and then n Square minus one can we make a comparison is this less than that I will tell you not clear maybe not true I don't know just no why not here's the thing if you look at this yes we can say
1:09:33
uh Square both sides first so I Square everybody first so x squared greater than N squared is greater than one squared that's okay and then minus one to everybody that's x squared minus 1 greater than N squared minus one greater than zero that's okay and then I will take the reciprocal
1:09:50
on this and that and that's okay and right here we have a greater than your both positive after the reciprocal this becomes less than that's okay but here's the thing on the top I have X on the top here we have n but X is greater than n ah the greater than and less than again they're fighting [Music] so I cannot just replace the X with I cannot just replace the X right here with I cannot just replace the one with X and then one with n and then keep the inequality that's not
1:10:26
clear that's no no good so let's go back to the idea that maybe you can somehow get rid of that X
1:10:39
howdo well on the bottom we have x squared minus one wouldn't it be nice if we have an X right here as well yes so that we can factor out the X earlier you don't want to factor out the x squared minus one right here it doesn't really do you any good but if you can somehow have x square minus X right here don't get fat or juice X and then we can cancel the X and it's much easier right because earlier as I told you the axon types are parting us
1:11:11
now can we compute this and that though yes check this out they both have x square
1:11:18
When U minus 1 here compare with minus X which one is bigger in fact -1 is bigger than minus X
1:11:32
yeah so the whole fraction here is smooth I'll show you real quick here we know X is bigger than one just compute X and one yeah and then you can square both sides this is x square
1:11:50
greater than one no I want to do it like this sorry a little bit higher
1:11:57
let's do this X is greater than one multiply negative and process or negative DX is less than negative U1 yep yes and then if I add x square on both sides x squared minus X is less than
1:12:18
x squared minus one yes and then if I take the reciprocal I will get 1 over x squared minus X being greater than 1 over x squared minus one and of course they're all positive because x squared is greater than 1 minus 1 is greater than um 0 and this right here will be greater than
1:12:50
zero as well so yeah they're all positive so this is okay and as you can see
1:12:55
this is the same as saying 1 over x squared minus 1 being less than 1 over x squared minus X all right so hopefully this is clear now that being said I can factor out the X
1:13:12
and then cancel this out and cancel this out perfect
1:13:18
yeah so let me actually put it down here this is less than original the X over x squared minus X but that's why I put on her year and if I taught the X I cancel this in that out so I can just move this to that then I will erase this because that's just some algebra work right so that's done so this thing is equal to 1 over x minus one okay now it's just this part right here the rest is pretty straightforward because X is greater than N I can replace this with it's just n
1:14:03
so sorry let me let me write it down again 1 over x minus 1. I'm going to replace the X with n
1:14:16
and because x minus 1 is bigger than n minus 1. so that means this whole fraction is less than that fraction now this expression is just with n so we can just put it on the side one over n minus 1 set it to be Epsilon we know 1 over Epsilon will be n minus 1 and will be 1 over Epsilon plus one so we'll come here and say and is one over Epsilon plus one put that here I'll just say this is equal to 1 over 1 over Epsilon plus one and then minus
1:14:55
one Auto Creator is equal to Epsilon okay done just like that perhaps I want to just be a little bit picky and to write down slightly better Organization for you guys
1:15:09
this is less than n minus one like this and with that this is equal to 1 over 1 over Epsilon plus one minus one which is just absolutely
1:15:27
prettier my Poor's not being enough otherwise I'll just write down everything in one line anyway that's it
1:15:48
[Music] okay number 16. limit as X approaching infinity x square plus 2X that will give us Infinity all right Infinity so proof given
1:16:09
this Infinity tells us that we should use Big M
1:16:14
now Infinity tells us so should you speak n so choose begin all right suppose
1:16:24
X is bigger than begin then we are going to check
1:16:30
our function is bigger than the Big M so we have 2X plus so 2 x squared plus 2X hopefully this is bigger than m [Music] all right what can we do well check this out X is bigger than n and we can safely say that they are both greater than zero x squared plus 2X you can ignore the 2X and you can say that's bigger than x square yes this inequality of the jet yes you can also
1:17:08
ignore the x square and just put on 2X is this inequality legit yes which one's easier well this one is easier so I'm just going to ignore the x square yeah this is in fact bigger than that but we're not 2 is bigger than one so how can one say this is bigger than just X
1:17:33
's because so this is bigger than n
1:17:39
and I want and to be bigger than that's just how to do what n is equal to M done this works because right here if n is equal to M I can say this is equal to m and done yeah
1:17:58
just like that it looks weird but it works it's legit this singing quality is true this inequality is true this is also true and of course Dusty thanks to my choice right n is equal to M and we're done if you really want to you could have done the S this is bigger than N squared plus 2 and and then set that to be
1:18:26
M and yeah if you do it like this then n will be the best end for this proof but you really don't need to this right here is okay and this right here is okay
1:18:40
remember in my previous video I mentioned it that you may have different choices different choices in terms of expression wise for Delta or for like uh the end stop right
1:18:55
okay number 17. limit as X approaching 1 x square plus 2X equals three so let's practice another quadratic one this is the Epsilon Delta definition same function like earlier so given Epsilon greater than zero choose Delta to be something so suppose
1:19:24
the distance between X and one is
1:19:28
between of zero and Delta check after the value of my function x squared plus 2X minus three
1:19:42
okay take a look right here can we Factor it yes we get what x minus 1 times X plus three this is less than Delta put that down right away this we need a number to work with right so go ahead and say Delta is equal to the minimum of 1 and something I don't know by looking at this right here we have absolute value of x is less than this one open this we get x minus 1 is in between of negative one and one and we want to go to
1:20:22
X plus three so let's just add three to all this no I forward to all that
1:20:33
four four four so this right here will be three this right here will be X plus three
1:20:42
oh yeah yeah yeah and this right here will be five then after five X plus three is less than five
1:20:52
okay so we can just say five so Delta should be Epsilon over five here so when you multiply by five they cancel come here put down Epsilon over five and
1:21:11
close that brace in there we'll write down less than or equal to [Music] [Music] so hopefully once you do these concussions like many times you'll see that oh yeah they are really really not that bad they're really really not that bad if you want to see some really bad ones you can check on my other videos for like proving the product of the limit is the limit of the product that was pretty bad
1:21:44
[Music] right number let's do another one-sided limit this is number 18. we are looking at the limit as X approaching 1 plus X over 2x minus 2. this will give you infinity
1:22:05
so here we go p f Infinity we use positive yeah given big amp greater than zero and you can imagine yourself I should also mention that if this was minus infinity you just choose you just say give a big M to be less than zero to begin with and later on just flip the new quality you can try download given begin greater than zero X is approaching one plus so we'll use Delta so choose Delta to be something suppose
1:22:44
the distance between X and one from the right hand side so no need for absolute value just x minus one in between of zero and Delta like so we want to show that this is going to be greater than M so hopefully X over 2x minus 2 is greater than m okay what should we do
1:23:13
check this out let me see then let me know if you think this makes sense for now all right X is in between of 1 and 1 plus Delta because you can just add one to everybody
1:23:26
X over 2x minus two we want to show that this is greater than M and that's what we want to show
1:23:34
this is a minus two can we just say that's greater than in order -2 like this because this denominator is smaller than that so the whole fraction is bigger than that
1:23:48
and so cancel one half
1:23:52
this right here is the jet don't don't give me wrong this right here is the chat but this is not what we are trying to do we are trying to make a connection between this and also the M all right so don't do it like this and that reason is because this time though what we have is x minus 1 this is pressure so we make sure we utilize that check this out we know x minus 1 is positive and C between of zero and Delta so right here can we just take the reciprocal on both sides
1:24:28
this and that so 1 over x minus 1 compare with 1 over Delta Philippine inequality that will be bigger yeah so you can factor out the two like this so we have X over and we can factor out two
1:24:47
let's let's put it down like this X over 2 times 1 over x minus 1.
1:24:57
we know this is greater than one over Delta slash okay so we can say 1 over Delta and we still have the X over 2 like so now check this out
1:25:10
we know this part
1:25:16
0 is less than x minus one meaning X will be greater than one yeah if x is greater than 1 can I just replace that with one like that yes 1 times 1 is 1 over 2 Delta it's legit okay now
1:25:39
it's just expression in terms of Delta so 1 over 2 Delta put it to be our Big M and just solve it so multiply Delta on both sides divided M on both sides Delta equals 1 over 2 m so I'll put that right here 1 over 2 m and then replace that right here that's an equal sign one over two times one over two M two cancel one over one over m equals capital M just m
1:26:14
yes number 18. let me check if it's still reporting very nice all right it's number 19 and number 18. moving down to number 19.
1:26:33
[Music] here we have the limit as X approaching Infinity x square over 3x squared plus 1.
1:26:45
1 third finite so we use Epsilon X is approaching Infinity used n so here p f given
1:26:57
Epsilon greater than zero choose and to be something suppose
1:27:06
X is a greater than n and then we want to show that the distance between our function
1:27:17
and the limit hopefully this is less than Epsilon all right so let's just work this out do the algebra I'll multiply the denominator so 3 times 3x squared plus one
1:27:33
and I'll do three times that which is three x square and I do this times negative one which is minus 3x squared one times that is just minus one still put that insert the absolute value cancel this cancel that absolute value of negative one is just one
1:27:51
and then better yet x square plus one order stuff this positive so you don't even have to worry about anything it's just always passing inside so you can take out the absolute value perfect
1:28:07
now do you want to replace the X with n if you want yeah but like I would just well wait what I wanted to do is I would say I'll take out the plus one and say that's less than 1 over 3 times that which is 9 x squared let's just just get rid of that one easier that way and one over nine is less than one yeah so I'm going to just say that's less than one over x squared okay now right here x is greater than n you can be sure you know it's going to happen it's good the
1:28:46
above positive and of course this means that I can replace the X with n and then we can say this is and square like this and this is still less than that to see clearly you can square both sides here x square is greater than N squared it's positive and then you can see that the reciprocal of them
1:29:14
like so told you so now once we have this go ahead put it somewhere down on the side one over n Square make it equal to Epsilon and solve it so we are talking about N squared equals 1 over Epsilon n equals 1 over
1:29:33
square root of Epsilon of course you just care for the positive version so go ahead and choose that and this is equal to 1 over 1 over square root of Epsilon square that is Epsilon and then
1:30:03
[Music] all right number 20. limit as X approaching 2 plus 1 over 3x minus six that will be positive Infinity
1:30:20
p f given M greater than zero
1:30:29
X is approaching two plus so I will just say we choose the altar to be something suppose
1:30:40
it's just one sided right so no need for absolute value x minus two it's in between of zero and Delta then we are going to check our function 1 over 3x minus six is bigger than m right and notice that x minus 2 here [Music]
1:31:04
well we better have the x minus two don't just ignore all that stuff right otherwise you can make the connection so I'm going to just factor out the three right here so we have one third and then we have one over x minus two well we can see that they are both positive if you take the reciprocal one over x minus two compare with one over Delta this is bigger so we can say this is bigger and
1:31:30
we have one over three times one over Delta yeah so this is pretty much one over three Delta put it to be Epsilon D Phi multiply Delta equals one over three Epsilon put it here one over three Epsilon this is equal to 1 over 3 and then we have one over three Epsilon
1:32:01
this is not cancel and one over one over Epsilon is equal to Epsilon done [Music]
1:32:20
who I'm glad that I'm not doing a bunch of this all right just and by the way if you know the reason for the number 24 leave a comment down below and appreciate that all right you should know the answer if you have been watching my videos for like a 24 something yeah okay let's do another quadratic one limit X approaching negative two and here we have x squared plus X Plus three this is equal to 5. all right Epsilon Delta
1:32:59
given Epsilon greater than zero choose Delta to be something and then suppose
1:33:11
the distance between X and -2 so it's become plus two like this
1:33:18
and then we have uh you know between zero and Delta check the distance between this and that so absolute value we have x squared plus X plus three minus five all right do the algebra so this is minus two so absolute value of x squared plus x minus two when we Factor this yes I will put down
1:33:48
X Plus two first and then x minus one this is less than Delta so we can put that down right away this is the case that we have to come here and write down minimum
1:34:06
so we can use more numbers right this implies perhaps the value of x plus 2 is less than one open this x minus 2 is in between of negative 1 and 1. and we're trying to get to what minus one so minus 3 to all this right here this right here gives you negative four and then x minus 1 and this is negative two and here's the reason why I want to do another case it's right here
1:34:40
we have negative 4 and negative two if I take the absolute value right here which number do I choose do I choose the negative 2 because it's bigger than negative four
1:34:53
that doesn't make sense because absolute value of forever is always positive [Music]
1:34:59
so what do I do you want to choose the number that has the bigger absolute value in this case is -4 and then take the optical for that so it's less than four
1:35:10
why I just realized how can show you guys why this works huh now I'll show you guys now when you apply the app store to this you see this is negative four this is negative two of course I can say that's less than four then if you look at this it's in between of negative four and four so I can say that's the value of x minus 1 is less than four
1:35:36
okay so that means Delta should be Epsilon over four Epsilon over four
1:35:48
close that and then we have the less than or equal to times 4 cancel and that's still that's Epsilon [Music] okay
1:36:15
four more four more am I talking about that was question number 21. three more three more all right three more number 22.
1:36:27
limit X approaching Infinity
1:36:32
and we have X over
1:36:36
2x minus eight and this equals 1 half here we go this is finite so we need Epsilon yeah you got it given absolute greater than zero and X is approaching Infinity so we need Big N yes you got it so choose big n now we want to assume suppose X is greater than big n and we want to see that the distance between this and I is less than Epsilon so absolute value X over two times
1:37:22
let me factor out the two for you guys have so 2 times x minus 4 minus one half like this all right the reason I want to do that is so that you see you don't need to multiply the two the lowest common denominator is just 2 times x minus four this x keep it and then is 2x times minus 1 so it's minus 2X 2 times negative 4 is negative eight yeah so it's negative 8 times negative one is positive eight [Music]
1:37:59
yeah add to the value all right so this is uh minus X
1:38:16
no no no no why am I talking about no no no what I'm talking about sorry all right I'm gonna just do this right here for you okay multiply x minus four this is what happens when you are kind of tired and they're trying to do a basic math all right they have the same denominator now so 2 times x minus four this is X distribute the negative one so it's minus X and the minus minus B counts plus
1:38:42
cancel cancel all right four is positive already doesn't matter oh you know what we can reduce this that's kind of cool for the first time so we just have a two right here over we have x minus 4. yeah 20 absolute value well you don't need to have absolute value if you come here and say that's greater than four will it happen
1:39:10
you'll see so that's what we have and now because X is greater than n I'm just going to compare that with 2 over the N minus 4. X is greater than n meaning x minus 4 is greater than M minus 4 and this will still be positive possibly positive so the whole fraction will be less than that right unless if you want you can you know do more work but I think this is also pretty clear
1:39:42
all right I'll do that for you guys real quick
1:39:46
so that so that means x minus 4 is greater than n minus 4 is greater than zero so 1 over x minus four one over and minus 4 will be less than yeah [Music] yeah and then still positive anyway this right here two over n minus four again once you have the expression terms such as n set it to be Epsilon divide and multiply this thing so we have uh 2 over Epsilon equals n minus four and will be 2 over Epsilon plus four put it here to over Epsilon plus four told you
1:40:33
yeah so this is equal to 2 over we have two over Epsilon plus four and after that we have the minus four work that out you will see this is going to be just Epsilon yeah two more
1:40:55
if you're still watching or maybe just skip the video to this point I don't know but if you're still watching if you have been watching all the way leave a comment down below let me know I really really appreciate that and you are fantastic seriously you're awesome you're phenomenal ah
1:41:17
[Music] all right number 23.
1:41:22
here limit X approaching 3 1 over parenthesis x minus three Square this is Infinity this is a two-sided limit but this is square so its Infinity is like going up yeah we have a 40 cross on top of street but here let's go ahead and do the proof given Big M because we have the infinity it's greater than zero X is approaching three so I'll choose what
1:41:53
Delta yes you got it choose Delta to be something we don't know yet suppose the distance between X and three is between 0 and Delta hopefully we can show that is bigger than M so check
1:42:12
our function 1 over x minus three Square [Music] check this out we have this thing right here right and this and now both positive we can just take the reciprocal so we are looking at one over x minus three and one over Delta and you will get greater than it's not exactly what we need yeah especially once we know they are both positive I'm not taking a reciprocal of the zero though I'm just saying they're both positive I can square this and square that and I can still
1:42:50
maintain the inequality so this is 1 over x minus 3 Square 1 over Delta Square I still maintain the inequality yeah so I can actually just say that's greater than 1 over Delta squared very nice huh
1:43:16
let's just put everything down in red
1:43:21
all right now in order to figure out 1 over Delta squared put it to pm Delta square equals 1 over m Delta equals 1 over square root of M that's what we want one over
1:43:40
square root of M so this equals 1 over 1 over square roots of M
1:43:48
and then we square that which is just begin
1:43:56
one more one more one more
1:44:05
yes I know I didn't do anything with like a negative Infinity you should try that
1:44:13
all right number 24 of course you should know this is to honor Kobe Bryant that's what 24. limit as X approaching Infinity
1:44:30
x square over X plus one this is infinity because the degree on the table is bigger than a degree on the bottom all right Infinity use M this infinitely tells you to use n so p f given
1:44:51
began greater than zero we are going to choose begin to be something suppose
1:45:01
X is greater than big n then we are going to check our function x square over X Plus 1 is bigger than begin
1:45:18
so what can we do can we just replace all the X's with n so n Square Over N plus 1 and say this is greater than that is this clear this is true what do you guys think [Music]
1:45:39
well let's just do everything detail to see if it is X is greater than n and that's just pretty greater than zero for now yeah and uh that means I can just add one to everybody so this means X plus one is greater than n plus one okay because you want to see that and take the reciprocal 1 over X plus one compare with one over M plus one
1:46:10
this is less than ta so
1:46:14
multiplied by x square n Square this inequality is now what we want already we are stuck
1:46:27
so no no
1:46:41
hey let's see earlier we said that okay X is greater than n greater than zero but why if we replace that with one and then we see that this is x squared of course that will imply that x squared is greater than 1 squared so can we say that's greater than one and over X plus one
1:47:05
yes but the thing is that if you replace this x with n then again the inequality will be fighting no no no no no no so you you cannot draw any kind of draw the conclusion that we want between this and this is actually smaller than that so no no whenever you write this down remember the requests you open the same way all the way and same in the same direction on the way so
1:47:37
kind of stuck so what do we do
1:47:45
well in this case I really cannot replace the x square with one square no but
1:47:53
let's check this out what if we can somehow
1:48:01
do what we did before remember there was a question I forgot the numbers that x square minus one and you compare that with x squared minus X and then we were able to factor the X and the reason is to carry off the X on the top yeah well let's try to do this here's the deal X is greater than one if I replace this one with X let's see what will happen x square over X Plus X
1:48:34
can we make a computer set
1:48:39
X is greater than 1 that means this fraction will so this denominator will be bigger than this denominator meaning that this will be smaller than that because well let me just say like this X Plus 1 this denominator is smaller than this denominator thus this whole fraction is bigger than this whole fraction all right and if you want to work it out of course that's totally okay so I'll just put this down on the side for you guys
1:49:15
we know X is greater than one so I can just add X on both sides of this
1:49:28
yeah and of course once you add the example the x is greater than once of course positive so I can just take the reciprocal now this is X Plus
1:49:39
X over I mean one on the top and all that one overheads plus one
1:49:46
yeah flipping equality becomes like this yeah so that means 1 over X plus one it's actually greater than 1 over X Plus X that's what we want and of course we can multiply x squared x squared and this inequality is still true so this right here is correct and this right here is correct so if you want to make it more clear maybe you want to put it on somewhere
1:50:16
all right so that's what we have and of course we can simplify this this is just x squared over 2X what's so good about this yeah cancel cancel in fact we're just looking at let me write it down x square over 2x cancel cancel so we are really just looking at X over two yeah
1:50:43
all right now we know X is greater than n so I will just replace this x with n and then we have that over two so that's also clear then what should we choose for n
1:50:58
go ahead and just write down n over to to be Big M and it should be 2m yeah so n is 2m so this n is equal to 2m and then after that divided by 2 cancel out ladies and gentlemen we get that Big M
1:51:20
yes so here you have it 24 rigorous limit proof and hopefully this right here helps you with your Calculus class or real analysis class proofs are not so bad just practice a little bit there are worse ones than this was anyway as always that's it [Music]
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