Question Number 42933 by Joel578 last updated on 05/Sep/18
$$\mathrm{Suppose}\:\mathrm{that}\:{f}\:\mathrm{and}\:{g}\:\mathrm{are}\:\mathrm{two}\:\mathrm{functions}\:\mathrm{such}\:\mathrm{that} \\ $$$$\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:{g}\left({x}\right)\:=\:\mathrm{0}\:\:\:\:\mathrm{and}\:\:\:\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:\frac{{f}\left({x}\right)}{{g}\left({x}\right)}\:\:\:\mathrm{exist}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:{f}\left({x}\right)\:=\:\mathrm{0} \\ $$
Commented by MrW3 last updated on 05/Sep/18
$${h}\left({x}\right)=\frac{{f}\left({x}\right)}{{g}\left({x}\right)} \\ $$$$\Rightarrow{f}\left({x}\right)={h}\left({x}\right){g}\left({x}\right) \\ $$$$\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:{f}\left({x}\right)=\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:{h}\left({x}\right)×\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:{g}\left({x}\right)={b}×\mathrm{0}=\mathrm{0} \\ $$