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We've just learned a bunch of laws for computing limits, but unfortunately usually limits are
not given exactly in the form that we could apply one of these laws. One example of a
kind of limit that you might see would be a limit that looks like this. Limit as X goes
to 2 of X squared minus 4 over X minus 2. In this limit, if we're trying to use the
limit laws, we would split it apart and we would have limit as X goes to 2 on the top
of X square minus 4, divided by the limit as X goes to 2 on the bottom of X minus 2.
And, each of these limits we can split apart further using the limit laws. If we do that,
we're going to see that we're going to have limit as X goes to 2 of X squared on top minus
limit as X goes to 2 of 4, divided by limit as X goes to 2 of X minus limit as X goes
to 2 of 2. And, I think that we can evaluate these limits at this point. So, the limit
as X goes to 2 of X squared is going to be 4, the limit as X goes to 2 of 4 -- well that's
just a constant, so it's 4. Divided by the limit as X goes to 2 of X is 2 minus the limit
as X goes to 2 of 2 -- that's a constant so it's going to be 2. So, what we get here is
zero over zero. So, zero over zero -- what does that mean? What is that telling us? Well,
first of all, is zero over zero a number? And the answer is no. We can't divide by zero,
right? That's not a number. So, does zero over zero mean the answer is undefined? Again,
the answer here is no. Now, up until this point in your life, zero over zero, or dividing
by zero, has always meant something was undefined. But now, we're doing limits. We're not plugging
in. So, if somebody were to ask me for the function F of X equals X squared minus 4 over
X minus 2, what is F of 2? I would definitely say it's undefined. But, if somebody asked
me, what is the limit as X approaches 2 of X squared minus 4 over X minus 2 -- that's
a trickier question! Because now we're asking what's happening around 2, not exactly at
2. Can we plug in points that are near 2 here? Yes, we can. So, we should be able to find
this limit. And, in fact, if we look at this function graphically, it turns out it looks
like a straight line with a hole in it at 2. And how would I know that? Well, X squared
minus 4 over X minus 2 can be simplified -- I can factor the top into X minus 2, X plus
2, divided by X minus 2. And now you can see the same factor in the top and the bottom.
And immediately you want to cancel those X minus 2's, right? You want to say that F of
X is equal to X plus 2. And it is, sort of. You see, this isn't really F of X, let's call
it G of X -- it's a new function. G of X is the function X plus 2 -- it's the function
that is exactly this line with no hole in it. But, since we have a hole in our function
at 2, F of X isn't exactly the same -- it has a different domain, different X values
you're allowed to plug into it. So, G of X is equal to F of X except when X equals 2.
What does that mean for us and our limit? Well, if we're trying to take the limit as
X goes to 2 of X squared minus 4 over X minus 2, we don't care about the point 2, right?
We don't care about what happens at 2, we care about what happens nearby. So, when doing
limits we are allowed to do this simplification of canceling a factor out of the top and the
bottom. So, we can say that the limit as X goes to 2 of our original function is the
same as the limit as X goes to 2 of our new, reduced function X plus 2. But that's a function
we can use the limit laws on. So, limit as X goes to 2 of X plus limit as X goes to 2
of 2, gives me 2 plus 2, which is 4. And this is kind of interesting, because F of 2 is
undefined, but the limit as X approaches 2 of F is 4. The main thing to take away from
here is you can factor and cancel terms when taking a limit. If you get a form of zero over zero, you want to cancel. So, depending
on the difficulty of the problem, it may be harder than just factoring something out and
canceling it. There may be more algebra involved. But, you always can find some way of canceling
if you're getting zero over zero. So, the moral here is we want to do more algebra.
And, as you continue on in the lesson, we'll see some of the more difficult types of algebra
that you might have to do.