[MUSIC] Limits are probably the most important concept in this course.
So we should really have a definition of what we mean by limit.
Now here is what we mean by limits. To say that the limit of f of x as x
approaches a is equal to L means that f of x can be as close to L as desired by
making x close enough to a. There is a tons of subtlety to this
definition so it's worth to look at an example.
So let's take a look at this function. This is the function that takes an input
x and spits out x^second minus one divided by by x minus one.
So let's try plugging in number three into this function.
So I plug in number three into this function and I have to just compute,
right? Three squared minus one over three minus
one well, that's three squared is nine minus one is eight three minus one is two
and nine divided by two is four And sure enough, out of this function comes the
number four Let's look at that example again but with a little bit more detail.
this is actually a pretty complicated function.
Alright? But I can open up the function.
Alright. And take a look at how the functions
actually doing its calculations. You can think of this function as having
three different steps. Alright.
One of the steps squares its input and subtracts one, and so I calculate the
numerator. Another step just subtracts one from its
input. The outputs of those two steps then get
plugged into the division. And that's how I get the output of this
big complicated function. Now, something like x^two - one,
you could also think of that as having some, you know separate steps as well.
But this is good for right now. Okay.
Now let's see what happens. I take the number three and I plug it
into the function. Alright?
Now I'm going to be calculating the numerator and the denominator separately,
so I'll take those 3s, and up here, I'll look at three^two - one and I'll get out
eight. And down here, three - one became two Now
the eight and the two get plugged into the division, and eight divided by two is
four and that becomes the output of the function,
right? Input's three, output is four but when I
look at it this way, I can see how all the steps are, are playing out.
Okay. I evaluate the function at three, but who
cares? Well, let's try to evaluate the function
at one instead of at three. So what happens when we plug in the
number one into this function? I got the number one here.
I'm going to look inside. I'm going to open up this function.
Now imagine I've got this number one. I'm going to plug it into the function.
All right. Now I'm going to be evaluating the
numerator and denominator separately, so I'm going to take this one and split it
up, and plug it into the numerator and the denominator.
The numerator sends its input to its input squared minus one.
So one^two minus one is zero and the same thing down here, one - one is zero Now
I've got 0 and 0 which I'm going to be plugging in to the.
Okay, very bad. Right?
I'm dividing by zero and I can not proceed, so this function is not defined
at one. So I can't plug one into the function.
But if I wanted to figure out what the function's value was that inputs near
one, I could do that. So let's try to plug in one point one
instead so let's plug one point one into this function.
I can't plug in one because I need to divide them by zero, but let's try
plugging in one point one I'm going to open up the function again and take one
point one plug it into the function. Now one point one is going to to be
evaluated in the numerator and the denominator.
one point one^second minus one is twentyone.
And one point one minus one became one. Now twentyone and one are going into the
division. And twentyone divided by one is two point
one So when I evaluate the function at one pint one I get out two point one.
Instead of just plugging in one value, let's plug in a whole bunch of values.
We'll make a table. So use that same function again.
F of x is x^second minus one divided by x minus one Now, I can't plug 1 into the
function, 'because if I plug in one, I'd be dividing by zero, and I can't divide
by zero. One isn't in the domain of this function.
But I can plug in numbers near one, right?
And we saw that one point one if I plug in that, I get two point one.
Right? And if I plug in one point zero one I get
two point zero one If I plug in 1 point zero zero one I get two point zero zero
one. Right?
And so on. If I plug in 1.000001 I get 2.000001.
Right. Well, what's going on here?
I could summarize this situation by saying the following.
The limit of x squared minus one over x minus one as x approaches one is equal to
two. Why is that?
Well, this is because. I can make x^2-1 over x minus one.
As close to two as I want. If.
I make x. Close enough.
To one. Lets see.
Here's my table alright if you want the output of this function to be within a
billionth of two all you need to do is to make sure that your input is within a
trillionth of one alright. As long as your input is close enough to
one you can guarantee that your output is as close to two as you like.
This is just looking at a table of values.
You know, maybe a dozen values and seeing what they're getting close to.
It would be a lot better if there were a more convincing argument.
So let's go back to our definition of limit.
To say the limit of f of x equals l means that f of x can be made as close to l as
you desire by making x close enough to a. And let me emphasize something.
Close enough. But not equal.
To a. Why does something like this matter?
Well, let's go back to our example. In our example the function wasn't
defined at one. But the limit doesn't depend upon the
function's value at one. It only depends on the function's value
near one. So x squared minus one over x minus one
is equal to x plus one as long as x isn't equal to one right.
As long as x isn't one this is a true statement.
So now what's the limit as x goes to one of x squared minus one over x - one.
Well, this is the limit as x approaches one of x.
one = one. because the limit doesn't depend upon the
value of the function at one. It only depends upon the values of the
function near one. And as a result, these two things have
the same limit. Even better the limit of x plus one, as x
approaches one, well that's the limit of a sum.
And the limit of a sum is the sum of the limits.
So I can rewrite this limit as the limit as x goes to one of x plus the limit as x
goes to one of one. And what the limit of x as x goes to one?
Well that's asking what can I make x close to if I make x close enough to one?
Well that's one. And the limit of one as x goes to one is
asking me what's one close to when x is close to well there's not even an x in
this right wiggling x doesn't affect this at all so that limits also one.
And one plus one is two so indeed the limit.
x^twenty two minus one divided by x - one as x approaches one is two.
Limits provide information about what a functions values are approaching,
alright. It's a way of accessing otherwise
forbidden information. I might not be able to plug in the value
one, because that would have entailed dividing by zero.
And yet I know, that the functions output is as close to two as I like.
As long as the input is close to but not equal to one.