Answer
a) not true
b) true
Work Step by Step
a) This is not true. For example $$ f(x)=\left\{ \begin{array}{ll} x & \mbox{if } x \neq 0 \\ 1 & \mbox{if } x= 0 \end{array} \right.$$
b) True, $$\lim\limits_{x \to 0}f(x)=\left(\lim\limits_{x \to 0}x\right) \left( \lim\limits_{x \to 0}\frac{f(x)}{x}\right)=(0)(1)=0.$$
