[MUSIC]. A lot of calculus is about understanding
the qualitative features of functions. Nobody really cares about f of 17.
But you might care about f of a big number, qualitatively.
We're going to try to make this concept of big number a bit more precise.
Well, here's how we're going to get around talking about a big number.
We're going to talk about the limit. The limit of f of x as x approaches
infinity equals to l. Means that f of x is as close as you want
it to be to l, provided x is large enough.
So instead of talking about just plugging in a big number, I'm going to say F at
some big number, so to speak, is equal to L if I can make the output of F close to
L by plugging in big enough numbers. At this point we've seen a bunch of
different definitions of limits. But they're all united by a common theme.
What if someone had asked us to cook up a definition of the limit of F of X equals
infinity as X goes to infinity? Could we come up with a definition for
this? Yeah, absolutely we can, here we go.
The limit of F of X as X approaches infinity equals infinity means that I can
make F of X as large as you want it to be provided X is large enough.
Consistently when we're talking about infinity in limits we're never actually
talking about a specific value. We're just talking about a value which is
as big as you want it to be. Let's go do an example at the blackboard.
There is a question. With the limit, of 2x over x+1 as x
approaches infinity. Before we dive into this analytically,
lets get some numeric evidence. This is my function, again fx)= of 2x
over x+1.1. I want to know qualitatively, what
happens when I plug in big numbers in this function?
So, lets say f100). of 100.
Well, that's not too hard to figure out. 22 times 100 is 200 and 100 plus one is
101. Now, 200 divided by 101 is pretty close
to 2. And there's nothing too special about
100, right? If I'd done this four million I would
have gotten two million over a million in one, which would be even closer to 2.
Numerically, it looks like this limit's two, but we just figured that out by
plugging in some big numbers. I want a more rigorous argument, some
analytic that is limit is actually equal to two.
How I'm going to proceed? My first guess would be to use the limit
law for quotients. This is the limit of a quotient which is
the quotient of limits provided the limits exist and the limit of the
denominator is none zero. Bad news here is the limits don't exist.
The limit of the numerator is infinity which isn't a number.
So I can't use my limit law for quotients here.
Instead I'm going to sneak up on this limit problem by wearing a disguise.
I'm going to multiply by a disguised version of one.
I'm going to multiply by one over x divided by one over x.
Now this is just one. Admittedly, I have changed the function.
The function's not defined anymore at zero.
But for large values of X, this doesn't affect anything.
And I'm taking the limit as X goes to infinity.
So I only care about agreement at large values of X.
I'll do some algebra. This limit has now the limit of 2X times
one over X. Which is two divided by the limit of X
plus one times one over X. Which is one plus one over X.
Maybe it doesn't look like I've made a lot of progress here, but this is a huge
progress. The limit of the numerator is now a
number, it's two. And the limit of the denominator is also
a number. And a non zero number, at that.
So I can use my limit law for quotients. This is the limit of a quotient, which is
the quotient of a limit. It's the limit of two, the numerator
divided by the limit of the denominator, as x approaches infinity.
The limit of the numerator is just the limit of a constant, which is 2.
The limit of the denominator is a limit of a sum, which is the sum of the limits
provided the limits exist, and they do. The limit of 1 is just 1.
And what's the limit of 1 over x as x approaches infinity?
Well that's asking, what is 1 over x close to when x is very large?
Well I can make 1 over x as close to zero as I like if I'm willing to make x large
enough. So the limit of 1 over x as x approaches
infinity is zero. That means my original limit is 2 over 1
plus zero, which is 2. [MUSIC].