Calculus - esplion-delta prove

Question

$$\lim_{x \to \infty} x\cos\frac{1}{x}=\infty$$

All $N>0$ exists $M>0$ so all $x>M$ appiles $x\cos\frac{1}{x}>N$

I am not really sure how to approach this, any help will be appreciated.

Answer 1

Hint:

If $x > 1$ then $\cos \frac{1}{x} > \frac{1}{2}$ so

$$x\cos\frac{1}{x} > \frac{x}{2} > \frac{M}{2}$$

So try with $M = \max\lbrace 1,2N\rbrace$