$\mathop {\lim }\limits_{x \to 0} {x^4}\cos \frac{1}{x} = ?$

The given limit $ = \mathop {\lim }\limits_{x \to 0} {x^4} \times \mathop {\lim }\limits_{x \to 0} \cos \frac{1}{x}$

$ = 0 \times $ a finite number

(cosine function is a bounded function lying between -1 to 1)

= 0