Answer
To prove the limit, prove that given $\epsilon\gt0$ there exists a $\delta\gt0$ such that for all $x$ $$0\lt|x|\lt\delta\Rightarrow|f(x)|\lt\epsilon$$
Work Step by Step
$$\lim_{x\to0}x\sin\frac{1}{x}=0$$
Our job is to prove that given $\epsilon\gt0$ there exists a $\delta\gt0$ such that for all $x$ $$0\lt|x|\lt\delta\Rightarrow|f(x)|\lt\epsilon$$
1) We know that for all $x$, $$|\sin\frac{1}{x}|\le1$$
Therefore, $$|x\sin\frac{1}{x}|\le|x|$$
2) Now let an arbitrary value of $\epsilon\gt0$, and let a value of $\delta$ so that $\delta=\epsilon$.
For all $x$, as $0\lt|x|\lt\delta$, we have $$|f(x)|=|x\sin\frac{1}{x}|\le|x|\lt\delta$$
$$|f(x)|\lt\delta$$
Yet since $\delta=\epsilon$, $$|f(x)|\lt\epsilon$$
The proof has been completed.
