Proving if a function is continuous or not

Question

Can somebody explain how to solve this problem:

Answer 1

The first question is asking you to show that $$\lim_{x \to 0} x \sin \frac{1}{x} = 0.$$

Hint:

Use the fact that $-1 \le \sin \frac{1}{x} \le 1$ for all $x \ne 0$.

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For the second question, you just need to determine whether $$\lim_{x \to 0} \sin \frac{1}{x}$$ is equal to $0$ or not.

Answer 2

Hint: $|f(x)-f(0)|=\left|x\sin\frac{1}{x}\right|\leq |x|=|x-0|.$

While as $\sin\frac{1}{x}$ oscillates as $x\rightarrow 0.$

Answer 3

Let $\epsilon >0$ Clearly $|f(x)-f(0)|<x$ for all $x$ if we set $\delta=\epsilon$ we obtain $|x-0|=|x|< \delta \Rightarrow |f(x)-f(0)|<x$