Evaluate $\lim_{x\to 0} \sin (\frac {1}{x})$

Question

Evaluate:

$$\lim_{x\to 0} \sin (\dfrac {1}{x})$$

My Attempt:

let $$\dfrac {1}{x}=k$$ As $\dfrac {1}{x}\to 0$, $k\to \infty$ Now, $$=\lim_{k\to \infty} \sin k$$

Answer 1

HINT: you can consider the sequences $x_n = \frac{2}{n\pi}$, or $x_n=\frac{2}{4(n+1)\pi}$, or $x_n=\frac{2}{4(n-1)\pi}$ when $n\to \infty$.

Answer 2

You are correct in your reasoning thus far. You can observe that the limit of $\sin x$ as $x$ approaches infinity doesn't exist, and neither does this limit.