finding points of discontinuity of the function $f(x) = \frac{x}{\sin x} $.

Question

finding points of discontinuity of the function $f(x) = \frac{x}{\sin x} $.

My answer is: The point of discontinuity is $0$ and it is a removable discontinuity because the limit is 1, am I correct?

Answer 1

Note that we expect a discontinuity everytime the denominator, i.e., $\sin x$, is $0$. This occurs when $x=n\pi$ for some $n\in\mathbb Z$.

Now note that when $n\neq0$, the numerator of the function will approach $n\pi$ while the denominator will approach $0$. Hence, these are not removable discontinuities but rather vertical asymptotes.

On the other hand, when $n=0$, then $\lim_{x\to0}\frac x{\sin x} = 1$, so this is a removable discontinuity.