# How do you find lim_(x to 0) xcos(1/x)?

Aug 28, 2017

Given: ${\lim}_{x \to 0} x \cos \left(\frac{1}{x}\right)$

We know that the cosine function is bounded by -1 and 1:

$- 1 < {\lim}_{x \to 0} \cos \left(\frac{1}{x}\right) < 1$

And we know that

${\lim}_{x \to 0} x = 0$

Therefore, the limit of a bounded function mulitplied by a function that goes to 0 must be 0:

${\lim}_{x \to 0} x \cos \left(\frac{1}{x}\right) = 0$