# What is the limit as x approaches 0 of sin(1/x)?

Jan 5, 2015

This limit does not exist, or with other words, it diverges.

You can see this by substituting $u = \frac{1}{x}$. Then, as $x$ approaches zero, $u$ approaches infinity.
Therefore:
${\lim}_{x \to 0} \sin \left(\frac{1}{x}\right) = {\lim}_{u \to \infty} \sin \left(u\right)$
This limit does not exist, for the sine is a periodic fluctuating function.