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0:00
We're now going to study the concept of limits in calculus. Now limits are very
0:05
important in calculus because they're used to define and evaluate the derivatives of
0:11
functions. And the derivative is really the central theme in calculus. So to
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properly understand what a derivative is, we have to first understand what a limit is.
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So let's start with this example. So we're going to consider this function,
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f of x equals x squared, and we're going to ask this question. What happens
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to the values of the function f of x as the input values x
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are taken closer and closer to the number 2 without ever actually
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taking on the value of 2 itself? So one way to answer this question
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is to create a table as we have down here. Now notice in this table
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we have x values like 1 .8, 1 .9, 1 .99,
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1 .999. Notice that these numbers are getting closer to 2 from the
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lower side of 2 or from the left we say. And then we also have
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values like 2 .2, 2 .1, 2 .01, 2
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.001. These values are getting closer to 2 from the
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upper side of 2 or from the right are getting closer to 2. side of
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2. Now down here we have our function values for each of the x
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values in the top part of the table. So notice if we put 1 .8
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in for x and square it, we'll get back 3 .24. For
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1 .9, the function value is 3 .61 and so on. So if we look
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at these values, 3 .24, 3 .61, 3 .96, 3
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.996, notice it looks like they are getting closer and closer to the
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number 4. But we also have to check and see that the same thing is
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happening as x approaches 2 from the right. Well, if we look at the function
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values there, notice we have 4 .84. I'll use there. Notice we have 4 .84,
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4 .41, 4 .04, 4 .004. Yes, it
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looks like they are also getting closer to the number 4. So from
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this table, it appears that as x gets closer and closer to the number 2,
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written like this, x with that little right arrow, 2, the function values
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f of x are getting closer to the number 4. and we'll indicate that like
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this sometimes. Now we can also use the limit notation to indicate what's
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happening here. So in this notation right here, we use,
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and it's read the limit as x approaches 2 of f of x
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equals 4. Or sometimes we can write this as well. Now we'll write the
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limit as x approaches 2 of x. squared, since our function is x squared in
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this case, equals 4. So this is limit notation here, and we'll use this to
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indicate what is happening to a function f of x as the x values are
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getting closer and closer to some number c. Now in this case, we can
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also see this graphically instead of using a table.
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Now here we have the graph of the function f of x equals x squared.
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It's this blue curve right here. And we have
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our x values on the x -axis right here. Currently our x value is 2
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.8 and the corresponding function value is 7 .84. I think function value is 7
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.84. Now notice what happens to the function values here on the
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y -axis. As the x values are taking closer and closer to 2 from
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either side, you'll see that these values will be approaching the limiting value of 4.
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So if we do that, so you see the
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x values are getting closer to 2, And keep an eye on those y values
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on the y axis. It does look like and keep an eye on those y
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-axis. they're getting closer to 4.
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I can step that in a little more slowly. So you see currently x is
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2 .02. The function value is 4 .08. And then we
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eventually land right on 4. And the same thing
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if we let x approach 2 from the left -hand
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side, what we did previously was let x approach 2 from the right -hand side.
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Now, if we let the x values approach 2 from the left -hand side, these
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function values here on the y -axis will also approach 4 from below.
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So just keep an eye on those function values on the y -axis. Notice as
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x gets closer to 2 from the left, they are getting closer to 4. Notice
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as x gets closer to 2 from the left, they are getting closer.
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And then we land right on 4. So that illustrates what's going on graphically.
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So keep in mind graphically here. When you're thinking of
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limits graphically, you should think of the x values approaching a number along
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the x -axis. And you're looking at what's happening to the function values along. the
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y -axis. Are they approaching a particular number? Now, in this
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case, for this function f of x equals x squared, it's a very well -behaved
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function. So, you might have noticed that if we just plug the number 2
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in for x into our function right here, if we put 2 in here for
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x, we just get back 4, which is our limiting value. back four, which is
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our limiting value. And that is true for most well -behaved functions. And that is
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true. This is a nice polynomial.
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Polynomials are well -behaved functions, and you could just plug the number two into this
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function and get your limiting value. That's true for well -behaved functions. But,
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as we'll soon see, that's not always going to be the case. Now, here's a
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little note down here about how you can create a table on the TI -89.
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graphing calculator and the TI -83. On the
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TI -83 calculator we can just store the function x squared into the calculator
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and then have the calculator evaluated like this. So if we just press y equals
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we'll store x squared into the y1 register of our
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calculator just by hitting x. That's the x button right there. and we'll square it
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then we can go back to the home screen second
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and we want the quit right there now we can press vars
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go over to Y vars select number one for function
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and number one again for Y1 and then just evaluate that at
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1 .8 for example close off your parentheses and
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hit enter and if you want to redo that instead of pressing VARS again
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you can there's a little entry a yellow entry right there above the enter button
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second entry and then you can change the 8 to a 9
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for example hit enter again do that again change
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the nine to a nine nine one point nine nine and so on
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and so you see these 9 to a nine nine one point nine nine numbers
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are getting closer to four you can also do this on the 89 the TI
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-89 is a little bit easier so here's a TI -89 you can do the
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same thing
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clear that out store x squared into the y1 register x
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raised to the 2 power and then just go back to the home
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screen and evaluate why now you have to type
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y1 of 1 .8
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close off the parentheses hit enter and there you go 3
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.24 and then you can change the 1 .8 to a 9,
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and then a 9 .9, and so
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on. And you see these numbers are getting closer to 4, just like on the
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83. So that's how you can create these tables quickly using a graphing
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calculator like a TI -83 or 89.
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So based on our previous discussion, we have what we call an... intuitive definition
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for a limit, and it goes like this. If there exists a number L,
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such that the function f of x gets closer and closer to this
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single number L, now that's an important point, it has to be a single number
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L, as x is taken closer and closer to the number C
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without ever actually taking on the value of C. So x just gets
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very close to it, but it never actually reaches c. Then we say the limit
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of the function f of x is the number l as x approaches the number
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c, and we write this. So that's
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intuitively what we mean by as x gets closer to c, f of x
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gets closer to the number l. Now we'll make this definition more precise in
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an upcoming lesson and it's also important to note as I said that it has
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to be a single number L. Now as we'll discuss later
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and eventually prove, for well behaved functions like polynomials,
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rational functions, trigonometric functions, radical functions,
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and their combinations. Usually to evaluate a limit as x approaches c
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of f of x, you can just plug c in for x and evaluate f
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of c. That's for most well -behaved functions when f of c
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exists, of course. However, when f of c does
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not exist, then the limit as x approaches c, of f of x, it
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may or may not exist, so more detailed analysis of the problem will be required.
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x approaches c of f of x, it may
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And to illustrate this point, let's consider the next example.
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In this example, we want to evaluate the limit, as x approaches 1,
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of x cubed minus 1 divided by x minus 1.
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Now you notice in this case, if we put 1 in for the x in
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the numerator and in the denominator, we get 0 divided by 0.
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Now this is an undefined quantity, and whenever you see 0 over 0
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appearing in a limit, that indicates that the limit may exist. So
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more detailed analysis is going to be required. So for this particular problem,
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we're going to create a table and we'll also look at it graphically to try
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to evaluate the limit. So here we have a table. So we have our x
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values. Notice we have values like 0 .8, 0 .9, 0 .99, 0
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.999. They're getting closer to 1 from the left.
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And then we have 1 .2, 1 .1, 1 .01, 1
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.001. they're getting closer to 1 from the right. And let's see what the function
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values are doing. Now down here you see we have our function values. These numbers,
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like 2 .44, are obtained by putting 0 .8 in for x
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into the function. You can evaluate that, and you'll see that you get 2
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.44. Well, notice that as x approaches 1 from the left.
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side, it looks like these values are getting closer to 3.
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But we have to see if the same thing is true from the right in
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order for the limit to exist. We must be approaching the same number from the
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right as well for the limit to exist. Well, we see that that is the
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case. If we put 1 .2 in for x, we get 3 .64,
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3 .31, 3 .03. 0 .64, 3 .31, 3 .03, and then 3 .003 for
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our function values for these values of x. And it does look like these are
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getting closer to 3 as well. So it looks like we would state that our
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limit is 3. So for our answer up here from this table,
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it looks like we would write that this is equal to 3 based on this
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table. Now we can also see that graphically if we look at this graph down
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here. The blue curve here is the graph of x cubed minus
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1 divided by x minus 1, and keep an eye on what happens
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to the values on the y -axis, these function values right here,
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as the numbers here are taken closer to 1 from the left and from the
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right. So keep an eye on these function values right here on the y -axis.
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So it does look like, step that in a little more slowly, it does look
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like as x is approaching 1 from the left, the function values you see are
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getting closer to 3 on the y -axis. And the same thing if we approach
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1 from the right side. It looks like those function values are getting closer
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to 3 as well. So no matter how we approach the number 1, it looks
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like our function values are getting closer to 3. And so that's what we would
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declare this limit to be. Now let's just note one
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other thing here. When we're evaluating the limit of a rational function like this one,
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this is a rational function because it's the quotient of two polynomials. When you evaluate
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the limit as x approaches c, if you put c in for the x's and
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you get zero divided by zero, that indicates that the limit may exist,
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and so further analysis is required. Now we will look at techniques for
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proving. that this limit is 3 later on. If you have a
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rational function and you plug the c value in for x and you get something
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like 2 divided by 0 or any non -zero number divided by
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0, that indicates that your limit does not exist. So
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we would just write dne in that case. And we will look at this
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in more detail in another video as well. Now let's take a look at another
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example.
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Suppose we want to evaluate the limit as x approaches 0 of sin x divided
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by x. Now notice also in this case, if we try to plug 0
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in for the x, we get... the sine of 0 divided by 0,
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and remember the sine of 0 is 0, so we get 0 divided by 0
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again. And that indicates that our limit may exist, so we're going to have to
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look at this in more detail. Once again, remember, if we had gotten something like
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2 divided by 0, a non -zero number divided by 0, that would indicate that
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our limit does not exist, and will not exist. zero, that would indicate that our
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limit does not exist. And we'll look at that question in more detail soon.
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So what we want to do here is we'll create a table again and we'll
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also look at it graphically. So here we have a table. Notice we have x
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values approaching zero from the left and we have x
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values approaching zero from the right. And we want to see what our function
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values are approaching. Well, down here are the function values, 0
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.993, 0 .9983, and so on. Notice that it looks like
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from the left, the function values are approaching 1, but we also need to
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see what's going on from the right. Well, it does look like from the right,
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these function values are also approaching 1. So
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no matter how x approaches 0, our function values are approaching 1.
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And so it looks like this limit up here from the table is 1. And
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that's what we would write. Limit as x approaches 0, sine x over x is
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1. Down here we can see that graphically.
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Notice if we keep an eye on these x values down here
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as they approach zero from the right and from the left let's x values down
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here as they approach 0 from the see what is happening
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to our function values here on the y -axis see if they are approaching one
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graphically so let's let x approach zero here
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and keep an eye on the y -axis values
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so as x approaches zero notice those functions
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on the y -axis they do look like they are approaching
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1. Step that in a little faster, there we go.
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Okay, so we see the x values are getting close to 0 now, and the
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function values are getting close to 1.
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And there we go. x is almost 0 and the function values are very close
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to 1 and the same So, x is almost 0, and the function values are
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getting close to 1. thing from the
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left hand side as x approaches 0 from the left side notice those function values
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are still getting closer to 1 and so that indicates that our limit
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should be 1 as we have up here now also notice one other thing
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about this graph. And this was true about our previous graph as well. this graph
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and this was true about our previous Notice how we have a little
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opening right here, a little circle. That indicates, of course, that our function does not
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exist when we put 0 in for x. So our graph looks
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fine, except for right there when x is equal to 0, we have to have
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a little open circle on the graph of our function to indicate that it does
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not exist when x is equal to 0. However, the limit still
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does exist because these values here on the y -axis are getting
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closer and closer to 1 as x gets closer to 0,
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as we saw. So that concludes this introduction to limits.
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Thanks for joining me, and I hope you enjoyed this video.
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