Chapter 1 - Functions and Limits - 1.6 Calculating Limits Using the Limit Laws - 1.6 Exercises - Page 71: 39

$\lim\limits_{x \to 0}x^{4}cos \frac{2}{x}=0$

$\lim\limits_{x \to 0}x^{4}cos \frac{2}{x}$ $-1\leq cos \frac{2}{x}\leq1$ therefore $-x^{4}\leq x^{4} cos \frac{2}{x}\leq x^{4} $ and $\lim\limits_{x \to 0}x^{4}=0$ $\lim\limits_{x \to 0}-x^{4}=0$ Therefore by the squeeze theorem : $\lim\limits_{x \to 0}x^{4}cos \frac{2}{x}=0$