use the squeeze theorem to evaluate the limit: $lim_{x\rightarrow0}xcos(2/x)=$

Explanation:

Step1: Recall cosine range

We know that $- 1\leqslant\cos\left(\frac{2}{x}\right)\leqslant1$ for all $x\neq0$.

Step2: Multiply by x

Multiply each part of the inequality by $x$ (when $x\to0$, we consider both positive - and negative - valued $x$). When $x>0$, we have $-x\leqslant x\cos\left(\frac{2}{x}\right)\leqslant x$. When $x < 0$, the inequality signs flip, but still $\lim_{x\to0}(-x)=\lim_{x\to0}x = 0$.

Step3: Apply Squeeze Theorem

Since $\lim_{x\to0}(-x)=0$ and $\lim_{x\to0}x = 0$, by the Squeeze Theorem, $\lim_{x\to0}x\cos\left(\frac{2}{x}\right)=0$.

Answer:

$0$