Answer
Apply the squeeze theorem, we can prove that $\lim\limits_{x\to0}x^4\cos\frac{2}{x}=0$
Work Step by Step
We know that
$-1\leq\cos\frac{2}{x}\leq1$
Multiply by $x^4$ throughout,
$-x^4\leq x^4\cos\frac{2}{x}\leq x^4$
(the inequality direction remains, because $x^4\geq0$ for $\forall x\in R$)
Since $\lim\limits_{x\to0}x^4=0^4=0$ and $\lim\limits_{x\to0}-x^4=-0^4=0$
Therefore, applying the squeeze theorem, we have
$\lim\limits_{x\to0}x^4\cos\frac{2}{x}=0$
