Prove or disprove that $\lim_{|z|\to \infty}\sin \frac{\pi}{z}=0$

Question

Prove or disprove that $\lim_{|z|\to \infty}\sin \frac{\pi}{z}=0$

I know that $|\sin \frac{\pi}{z}|=|\frac{e^{\frac{i\pi}{z}}-e^{-\frac{i\pi}{z}}}{2i}|=|\frac{e^{\frac{i\pi}{z}}-e^{-\frac{i\pi}{z}}}{2}|$. How do I manipulate this expression. If it would be the case of real valued function, proof is simple.

Answer 1

You don't need to manipulate anymore $e^\frac{2\pi i}{z}$ is a continuous function with $\lim_{|z|\rightarrow\infty} e^\frac{2\pi i}{z}=1$ so you have

$$\left|\sin \frac{\pi}{z}\right|=\left|\frac{e^{{i\pi}/{z}}-e^{-{i\pi}/{z}}}{2i}\right|=\left|\frac{e^{{i\pi}/{z}}-e^{-{i\pi}/{z}}}{2}\right|\rightarrow\left|\frac{1-1}{2}\right|=0$$

Answer 2

Asserting that $\lim_{\lvert z\rvert\to\infty}\left\lvert\sin\left(\frac\pi z\right)\right\rvert=0$ is the same thing as asserting that $\lim_{z\to0}\left\lvert\sin\left(\pi z\right)\right\rvert=0$. It is clear that this holds, since $z\mapsto\sin(\pi z)$ is continuous and it takes the value $0$ at $0$.