Is Squeeze Theorem Relevant Here?

The function $f(x)=\sin(1/x)$ oscillates rapidly from $-1$ to $+1$ as $x\rightarrow 0$ , and the oscillation is too much for the function to have a limit(see the image).

Graph of the function y=sin(1/x)

This type of discontinuity is sometimes referred to as oscillating discontinuity , according to which a function seems to be approaching two different values, and in this case it seems to be approaching $1$ or $-1$ . Thus the limit doesn't exsist.

Let $x_k = \frac {-1}{2\pi \cdot k}$ when $k \rightarrow \infty$ ( $x \to 0^{-}$ ) and $k \in \mathbb{N}$ , then the limit above is 0.

Now, let $x_k = \frac {-1}{ \frac {\pi}{2} + 2\pi \cdot k}$ when $k \rightarrow \infty$ and $k \in \mathbb{N}$ , then the limit above is -1.

Therefore, the limit above doesn't exist due to definition of limit