Answer
a. 0
b. 0
Work Step by Step
(a) $\displaystyle \lim_{x\rightarrow 0}f(x)=\lim_{x\rightarrow 0}[f(x)\cdot\frac{x^{2}}{x^{2}}],\qquad (x\neq 0)$
$=\displaystyle \lim_{x\rightarrow 0}[\frac{f(x)}{x^{2}}\cdot x^{2}]\quad $... Law 4, product
$=\displaystyle \lim_{x\rightarrow 0}\frac{f(x)}{x^{2}}\cdot\lim_{x\rightarrow 0}x^{2}\quad $... Law 9. $\displaystyle \lim_{x\rightarrow a}x^{n}=a^{n}$
$=5\cdot 0=0$
(b) $\displaystyle \lim_{x\rightarrow 0}\frac{f(x)}{x}= \displaystyle \lim_{x\rightarrow 0}[\frac{f(x)}{x}\cdot\frac{x}{x}],\quad (x\neq 0)$
$=\displaystyle \lim_{x\rightarrow 0}[\frac{f(x)}{x^{2}}\cdot x]\quad $... Law 4, product
$=\displaystyle \lim_{x\rightarrow 0}\frac{f(x)}{x^{2}}\cdot\lim_{x\rightarrow 0}x\quad $... Law 8. $\displaystyle \lim_{x\rightarrow a}x=a$
$=5\cdot 0=0$
