$\lim_{x\to\infty} x\sin({1\over x})$

Question

Let $f(x)= x\sin({1\over x})\forall x>0$

Then evaluate

$\lim_{x\to\infty} x\sin({1\over x})$

My approach: $\sin({1\over x})$ is periodic function and $x$ denotes its amplitude. So I thought $f$ is an unbounded function and hence will not have any feasible value for $x\to\infty$

Answer 1

Notice

$$ x \sin ( \frac{1}{x} ) = \frac{ \sin \left( \frac{1}{x} \right) }{\frac{1}{x}} $$

Now, put $t = \frac{1}{x} $. Remember,

$$ \lim_{\alpha \to 0 } \frac{ \sin \alpha}{\alpha} = 1 $$