We want to solve limits that have the form nonzero over zero.

Let’s cut to the chase:

Which of the following limits are of the form ?

Let’s see what is going on with limits of the form . Consider the function While the does not exist, something can still be said. First note that as Moreover, as approaches :

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

Hence will become arbitrarily large, as we can see in the next graph.

PIC

We are now ready for our next definition.

Let’s consider a few more examples.

Here is our final example.

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.