Calculus: Problems and Solutions

Some functions don’t have a limit as the value of their parameter approaches . Here’s one example:

just doesn’t exist. Why is that? Because as increases, keeps oscillating between 1 and -1. It never approaches any one value, and so we can’t assign a limit to it.

Notice that this is different from a function having a limit of or . In that case, the limit exists, and is equal to plus or minus infinity. In the case above, the limit doesn’t even exist.

Here’s another example that sometimes trips students up: what is the following limit?