TITLE: Evaluate the limit at infinity
QUESTION [5 upvotes]: Evaluate: $$\lim_{x \rightarrow \infty}x\sin \frac{1}{x}$$

$$\lim_{x \rightarrow \infty}x \times\lim_{x \rightarrow \infty} \sin \frac{1}{x}=\infty \times0=\text{Undefined}$$
Is this the correct way to convey that the limit does not exist? Or is there a mathematical way to show that $$\lim_{x \rightarrow \infty} \sin \frac{1}{x} = 0$$
Other than just knowing that 1 divide by an infinitely large number approaches $0$.

REPLY [2 votes]: For $x$ big enough, $\tfrac1x$ is close to zero and $\sin \frac1x=\frac1x+O(\frac1{x^3})$. Then 
$$
x\sin\frac1x=1+O(\frac1{x^2})\xrightarrow[x\to\infty]{}0
$$