limits - Why does $limlimits_{xto0}sinleft(left|frac{1}{x}right|right)$ not exist?

Can someone explain, in simple terms, why the following limit doesn't exist?

$$\lim \limits_{x\to0}\sin\left(\left|\frac{1}{x}\right|\right)$$

The function is even, so the left hand limit must equal the right hand limit. Why does this limit not exist?

Answer

The function is indeed even, as we will see, this does not prove the existence of the limit.

Assume that the limit exists.

$$ \lim_{x\to 0^+} \sin\left(\frac1x \right) $$

Which is the same as your limit because the left-hand limit and right-hand limit will be equal (if the limit exists). Then since $x > 0$, I removed the absolute value sign.

$$ \lim_{u\to\infty} \sin u $$

Using $u = 1/x$, we see that the limit certainly does not exist.