U0S3V03 Limits and division.txt

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Let's talk some more about limits of quotients.
Previously, we said that if the limit of the numerator
was equal to L and the limit of the denominator was equal to M
and if M is not zero, then the limit of the quotient
is L over M. And that was part one for Limit Law for Division.
But what happens if the limit of the denominator is equal to 0?
The answer is, it's going to depend
on what the numerator is doing.
If the limit of the numerator is something other than 0,
then if you look at f of x over g of x, then when x
is approaching a, we're going to get
numerators that aren't near 0.
So they are not small.
But our denominators are going to be really close to 0.
So we're going to be dividing a number that's not small by one
that is really small.
And the result is going to be huge.
And as the denominator gets smaller and smaller,
this quotient is going to get even huger.
For instance, if we take 5 over 0.01, we get 500.
But if we have 5 over 0.001, we get 5,000.
Now we don't know the signs of these things.
So it could be huge positive or huge negative.
And if the denominator equals 0 exactly,
then this quotient might not be defined at all.
But one thing is for certain, as x approaches
a there's no way that this quotient can be approaching
any fixed number.
So the limit does not exist.

So that's part two of the Division Limit law.
But that still leaves a third case.
If the limit of the denominator equals
0 and the limit of the numerator also equals 0, then what?
And you might be tempted to say that this limit won't
exist either, but that's not necessarily so.
A lot of interesting things can happen.
For instance, let's say that we want the limit as x approaches
0 of 2x over x.
And here the numerator is continuous
so as x approaches 0, we can just plug in,
and we see that the numerator is approaching 0 and same thing
with the denominator.
But we don't want to say that this limit is 0/0.
0/0 isn't defined, but this quotient, I
mean we know it's 2.
Well, almost everywhere it's 2.
It's not equal to 2 when x is 0, but when
we're taking the limit as x approaches 0, what happens
at x equals 0 doesn't matter.
So this quotient is 2 everywhere that counts
and so its limit is 2.
OK.
You might be complaining that this
seems like an exception, sort of a special case.
But really this situation of a quotient where
both the top and the bottom go to 0, and yet
the limit of the quotient still exists,
that happens a lot in calculus, and it's incredibly important.
So we have some questions for you
so that you can dig in and really understand
how this comes about.
Talk to you later.