Question

Please help me with the below question.

Answer 1

The limit does not exist.

cos is continuous, so we have

$\displaystyle \lim_{x\to0} \cos\left(\frac1x\right) = \cos\left(\lim_{x\to0} \frac1x\right)$

but

• 1/x approaches +∞ as x approaches 0 from above

• 1/x approaches -∞ as x approaches 0 from below

hence the two-sided limit does not exist.