Answer
The limit is not defined.
Work Step by Step
We have to estimate the limit:
$\displaystyle\lim_{h\rightarrow 0} \cos\dfrac{1}{h}$
Compute $\cos\dfrac{1}{h}$ for values of $h$ close to 0:
$\cos\dfrac{1}{-0.01}\approx 0.86231887$
$\cos\dfrac{1}{-0.001}\approx 0.56237908$
$\cos\dfrac{1}{-0.0001}\approx -0.95215537$
$\cos\dfrac{1}{-0.00001}\approx -0.99936081$
$\cos\dfrac{1}{0.00001}\approx -0.99936081$
$\cos\dfrac{1}{0.0001}\approx -0.95215537$
$\cos\dfrac{1}{0.001}\approx 0.56237908$
$\cos\dfrac{1}{0.01}\approx 0.86231887$
Therefore we got:
$\displaystyle\lim_{h\rightarrow 0} \cos\dfrac{1}{h}$ does not exist.
