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[MUSIC] Limits are probably the most important concept in this course.

Â So we should really have a definition of what we mean by limit.

Â Now here is what we mean by limits. To say that the limit of f of x as x

Â approaches a is equal to L means that f of x can be as close to L as desired by

Â making x close enough to a. There is a tons of subtlety to this

Â definition so it's worth to look at an example.

Â So let's take a look at this function. This is the function that takes an input

Â x and spits out x^second minus one divided by by x minus one.

Â So let's try plugging in number three into this function.

Â So I plug in number three into this function and I have to just compute,

Â right? Three squared minus one over three minus

Â one well, that's three squared is nine minus one is eight three minus one is two

Â and nine divided by two is four And sure enough, out of this function comes the

Â number four Let's look at that example again but with a little bit more detail.

Â this is actually a pretty complicated function.

Â Alright? But I can open up the function.

Â Alright. And take a look at how the functions

Â actually doing its calculations. You can think of this function as having

Â three different steps. Alright.

Â One of the steps squares its input and subtracts one, and so I calculate the

Â numerator. Another step just subtracts one from its

Â input. The outputs of those two steps then get

Â plugged into the division. And that's how I get the output of this

Â big complicated function. Now, something like x^two - one,

Â you could also think of that as having some, you know separate steps as well.

Â But this is good for right now. Okay.

Â Now let's see what happens. I take the number three and I plug it

Â into the function. Alright?

Â Now I'm going to be calculating the numerator and the denominator separately,

Â so I'll take those 3s, and up here, I'll look at three^two - one and I'll get out

Â eight. And down here, three - one became two Now

Â the eight and the two get plugged into the division, and eight divided by two is

Â four and that becomes the output of the function,

Â right? Input's three, output is four but when I

Â look at it this way, I can see how all the steps are, are playing out.

Â Okay. I evaluate the function at three, but who

Â cares? Well, let's try to evaluate the function

Â at one instead of at three. So what happens when we plug in the

Â number one into this function? I got the number one here.

Â I'm going to look inside. I'm going to open up this function.

Â Now imagine I've got this number one. I'm going to plug it into the function.

Â All right. Now I'm going to be evaluating the

Â numerator and denominator separately, so I'm going to take this one and split it

Â up, and plug it into the numerator and the denominator.

Â The numerator sends its input to its input squared minus one.

Â So one^two minus one is zero and the same thing down here, one - one is zero Now

Â I've got 0 and 0 which I'm going to be plugging in to the.

Â Okay, very bad. Right?

Â I'm dividing by zero and I can not proceed, so this function is not defined

Â at one. So I can't plug one into the function.

Â But if I wanted to figure out what the function's value was that inputs near

Â one, I could do that. So let's try to plug in one point one

Â instead so let's plug one point one into this function.

Â I can't plug in one because I need to divide them by zero, but let's try

Â plugging in one point one I'm going to open up the function again and take one

Â point one plug it into the function. Now one point one is going to to be

Â evaluated in the numerator and the denominator.

Â one point one^second minus one is twentyone.

Â And one point one minus one became one. Now twentyone and one are going into the

Â division. And twentyone divided by one is two point

Â one So when I evaluate the function at one pint one I get out two point one.

Â Instead of just plugging in one value, let's plug in a whole bunch of values.

Â We'll make a table. So use that same function again.

Â F of x is x^second minus one divided by x minus one Now, I can't plug 1 into the

Â function, 'because if I plug in one, I'd be dividing by zero, and I can't divide

Â by zero. One isn't in the domain of this function.

Â But I can plug in numbers near one, right?

Â And we saw that one point one if I plug in that, I get two point one.

Â Right? And if I plug in one point zero one I get

Â two point zero one If I plug in 1 point zero zero one I get two point zero zero

Â one. Right?

Â And so on. If I plug in 1.000001 I get 2.000001.

Â Right. Well, what's going on here?

Â I could summarize this situation by saying the following.

Â 5:40

To one. Lets see.

Â Here's my table alright if you want the output of this function to be within a

Â billionth of two all you need to do is to make sure that your input is within a

Â trillionth of one alright. As long as your input is close enough to

Â one you can guarantee that your output is as close to two as you like.

Â This is just looking at a table of values.

Â You know, maybe a dozen values and seeing what they're getting close to.

Â It would be a lot better if there were a more convincing argument.

Â So let's go back to our definition of limit.

Â To say the limit of f of x equals l means that f of x can be made as close to l as

Â you desire by making x close enough to a. And let me emphasize something.

Â Close enough. But not equal.

Â To a. Why does something like this matter?

Â Well, let's go back to our example. In our example the function wasn't

Â defined at one. But the limit doesn't depend upon the

Â function's value at one. It only depends on the function's value

Â near one. So x squared minus one over x minus one

Â is equal to x plus one as long as x isn't equal to one right.

Â As long as x isn't one this is a true statement.

Â So now what's the limit as x goes to one of x squared minus one over x - one.

Â 7:13

Well, this is the limit as x approaches one of x.

Â one = one. because the limit doesn't depend upon the

Â value of the function at one. It only depends upon the values of the

Â function near one. And as a result, these two things have

Â the same limit. Even better the limit of x plus one, as x

Â approaches one, well that's the limit of a sum.

Â And the limit of a sum is the sum of the limits.

Â So I can rewrite this limit as the limit as x goes to one of x plus the limit as x

Â goes to one of one. And what the limit of x as x goes to one?

Â Well that's asking what can I make x close to if I make x close enough to one?

Â Well that's one. And the limit of one as x goes to one is

Â asking me what's one close to when x is close to well there's not even an x in

Â this right wiggling x doesn't affect this at all so that limits also one.

Â And one plus one is two so indeed the limit.

Â x^twenty two minus one divided by x - one as x approaches one is two.

Â Limits provide information about what a functions values are approaching,

Â alright. It's a way of accessing otherwise

Â forbidden information. I might not be able to plug in the value

Â one, because that would have entailed dividing by zero.

Â And yet I know, that the functions output is as close to two as I like.

Â As long as the input is close to but not equal to one.

Â