## Calculus 8th Edition

(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. $\displaystyle \lim_{x\rightarrow a}x=a$ $=5\cdot 0=0$