U0S3V11 Infinite Limits.txt

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We know that limits might not exist,
and they might not exist in a variety of ways.
It could blow up to infinity or it could blow up
to minus infinity.
Or there could be some other reason, like the function just
goes crazy.
So far, we've lumped all of these scenarios
under the label of DNE.
But sometimes, like when you're graphing a function,
you don't just want to know that a limit doesn't exist.
You want to know in what way it doesn't exist.
So that's what this short section is going to be about.
Let's take the function 1 over x.
And let's suppose we want its limit as x approaches 0
from the right.

Now, as x approaches 0, the denominator is approaching 0.
And this numerator, well, that's 1.
So it's not approaching 0.
And that means the second part of the division limit law
is going to tell us that this limit does not exist.
But does not exist in what way?
The limit law doesn't tell us that.
So let's investigate.
So our numerator is always 1.
And as x approaches 0 from the right,
the denominator is positive and getting really, really small.
So we're going to get things like 1 over 0.01, 1 over 0.001,
1 over 0.0001, et cetera.
And these fractions are 100, 1,000, 10,000.
So as x is going to 0 from the right, 1 over x is blowing up.
And its graph is going to be looking
like this, with this vertical asymptote at x equals 0.
So that's what's going on.
And when a function blows up like this,
we'll write that the limit is plus infinity.
And that just means that the limit does not
exist because it blows up in the positive direction, hence
the plus infinity.

If a function's limit doesn't exist because the function
blows up in the opposite direction,
then we'll say that the limit is minus infinity.
So we have some problems for you so
that you can play around and get comfortable with this.