What is the limits of this specific function?

Question

$$\lim_{x\to 0}\ x*[2sin\frac{1}{x}]=?$$

When I use the sandwich method, I get that the limit = 0. is that correct?

Answer 1

Rewrite as:

$x [Sin \frac{1}{x}]=\frac {[sin{\frac{1}{x}]}}{\frac{1}{x}}$

Using L'Hopital rule we get:

$\lim_{x→0} x [Sin \frac{1}{x}]= \lim_{x→0}[ Cos \frac{1}{x}] $

$\ Cos x∈ \{-1, 1\}$

Therefore:

$\lim_{x→0} [Cos \frac{1}{x}]=0 $