What can be said about limits that have the form nonzero over zero?

Let’s cut to the chase:

In our next example, let’s see what is going on with limits of the form .

- The numerator is positive.
- The denominator approaches zero and is positive.

Hence, the expression will become arbitrarily large as approaches . We can see this in the graph of .

We are now ready for our next definition.

**infinity**as goes to .

If grows arbitrarily large as approaches and is negative, we write and say that the
limit of is ** negative infinity** as goes to .

Note: Saying “the limit is equal to infinity” describes more precisely the behavior of the function near , than just saying ”the limit does not exist”.

Let’s consider a few more examples.

**determinate**, since it implies that the limit does not exist.

But, we can do better than that! As approaches :

- The numerator is a positivenegative number.
- The denominator is positivenegative and is approaching zero.

This means that

Canceling a factor of in the numerator and denominator means we can more easily check the behavior of this limit. As approaches from the right:

- The numerator is a positivenegative number.
- The denominator is positivenegative and approaching zero.

This means that

Here is our final example.

- The numerator is a negative number.
- The denominator is positive and approaching zero.

Hence our function is approaching from the right.

As approaches from the left,

- The numerator is negative.
- The denominator is negative and approaching zero.

Hence our function is approaching from the left. This means We can confirm our results of the previous two examples by looking at the graph of :

Some people worry that the mathematicians are passing into mysticism when we talk about infinity and negative infinity. However, when we write all we mean is that as approaches , becomes arbitrarily large and becomes arbitrarily large, with taking negative values.