U0S3V13 Infinite limits 2.txt

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You've looked at the functions 1 over x
and 1 over x squared as x approaches 0 from either side.
We saw that 1 over x has a limit of plus infinity
as you come in from the right.
And as you come from the left towards 0,
the limit is minus infinity.
So it was different on either side.
But with 1 over x squared, hopefully
you saw that coming in from the right
or from the left, either way the limit is plus infinity.
So these limits can be kind of delicate,
and just using the division limit law
isn't going to tell us everything we want to know.
Let's do a slightly more complicated example.
I'm going to take the function 3x over 4 minus x squared,
and let's look at the limit as x approaches 2.
Now as x approaches 2, the denominator is going towards 0
and the numerator is going towards 6, which is not 0.
So the second part of the division limit law
tells us that this limit does not exist.
And if we graph it, we can probably
expect a vertical asymptote here at x equals 2.
But we don't know if it's going to plus infinity
or minus infinity, and it might do something different
on either side.
So to figure it out, it's generally easiest
to deal with each side separately.
And let's just start with the limit
as x approaches 2 from the left, and we'll
need to pay very close attention to the signs of the numerator
and denominator.
This numerator is going to be near 6, which is positive.
The denominator we know is going to be close to 0,
but close to 0 how?
Well, when x is approaching 2 from the left,
that means x is going to be slightly less than 2.
So x squared is going to be slightly less than 4.
So our denominator is going to be just a little bit greater
than 0.
So we've got a numerator that's close to 6
divided by a denominator which is getting really close to 0
but positive.
So these quotients then are going
to be really large and positive.
So our limit from the left is blowing up
towards plus infinity, and the graph just to the left of 2
will look something like this.

If we look at the limit from the right,
well, we'll do something similar.
Our numerator is again going to be near 6.
Our denominator will still be near 0.
But this time we're looking at x's which
are slightly greater than 2.
x squared will be slightly bigger than 4.
And so 4 minus x squared is just a shade under 0.
So numerator near 6.
Got a denominator which is small but negative.
That means the quotient is large and negative.
And this limit from the right is then minus infinity,
and the graph is going to look like this just
to the right of 2.

This is the kind of stuff that you'll
have to do in order to figure out limits
where the function's output is going
to plus or minus infinity.
Eventually we'll discuss limits where
the input is going to plus or minus infinity,
but that'll be a lot later in the course,
and it's much easier to deal with those
once you know about derivatives.
And derivatives are what's coming up next.