Calculus: Problems and Solutions

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Limits That Don’t Exist

September 11, 2009 · 4 Comments

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

\displaystyle\lim_{x\rightarrow\infty} \sin x

just doesn’t exist. Why is that? Because as x increases, \sin x 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 \infty or -\infty. 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?

\displaystyle \lim_{t\rightarrow\infty} x\cos(\pi x)

Categories: calculus

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