[Math] Proving that $\lim\limits_{x\to 0} \sin(x)\cos(\tfrac1x) = 0$

Question

Prove that: $$\lim_{x\to 0} \sin(x)\cos(\tfrac1x) = 0$$

I am completely confused in knowing where to begin problems like this.

I have the beginning:
For all $\epsilon > 0$ choose $\delta = ?$…
When $|x-0|<\delta$ then $|f(x)-0| = |\sin(x)\cos(1/x)-0| \quad \ldots \quad < \epsilon$.

I need step by step help because I am completely lost in understanding how to complete this type of problem.

Answer 1

This is quite straightforward:

• $|\sin(x) \cdot \cos(1/x)| = |\sin(x)|\cdot |\cos(1/x)|\le |\sin(x)|$ for all $x \ne 0$.
• $|\sin(x)|$ tends to zero as $x$ tends to zero.
• By the Sandwich Theorem, $|\sin(x) \cdot \cos(1/x)|$ tends to zero as $x$ tends to zero.