The limit does not exist.
cos is continuous, so we have
[tex]\displaystyle \lim_{x\to0} \cos\left(\frac1x\right) = \cos\left(\lim_{x\to0} \frac1x\right)[/tex]
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.
