Answer
a. 0, b. see explanations.
Work Step by Step
(a) See graph, $\lim\limits_{x\to 0}g(x)=0$
(b) $\lim\limits_{x\to 0}g(x)=\lim\limits_{x\to 0}(x\cdot sin(\frac{1}{x}))=\lim\limits_{x\to 0}x\cdot \lim\limits_{x\to 0}sin(\frac{1}{x})$. We know that $ \lim\limits_{x\to 0}sin(\frac{1}{x})$ does not exit but the value oscillates within $[-1,1]$. Thus $\lim\limits_{x\to 0}g(x)=\lim\limits_{x\to 0}x\cdot \lim\limits_{x\to 0}sin(\frac{1}{x})=0\cdot \lim\limits_{x\to 0}sin(\frac{1}{x})=0$
