Question

Determine the limit when x goes to 0 of (sin(1/x)) if it exists.

Answer 1

The limit does not exist.

$h=\frac 1x, ~x\rightarrow0,~h\rightarrow \infin,$

$\lim\limits_{x\rightarrow0}sin\frac 1x=\lim\limits_{h\rightarrow\infin}sin(h).$

As $h$ gets bigger, $sin(h)$ keeps fluctuating between -1

 and 1. It never tends towards anything, or stops fluctuating at any point.