Question Number 72905 by Tony Lin last updated on 04/Nov/19
$${f}\left({x}\right)\geqslant\mathrm{0},\:{and}\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}{f}\left({x}\right)=\mathrm{0},\underset{{x}\rightarrow{a}} {\mathrm{lim}}{g}\left({x}\right)=\infty \\ $$$${then}\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}{f}\left({x}\right)^{{g}\left({x}\right)} =? \\ $$
Commented by mathmax by abdo last updated on 04/Nov/19
$${we}\:{have}\:{by}\:{taylor}\:{f}\left({x}\right)={f}\left({a}\right)+\left({x}−{a}\right){f}^{'} \left({a}\right)+{o}\left({x}−{a}\right) \\ $$$$\Rightarrow{f}\left({x}\right)\sim\left({x}−{a}\right){f}^{'} \left({a}\right)\:\:{because}\:{lim}_{{x}\rightarrow{a}} {f}\left({x}\right)=\mathrm{0} \\ $$$$\left({we}\:{suppose}\:{f}\:{derivable}\:{at}\:{point}\:{a}\right)\:\Rightarrow \\ $$$${f}\left({x}\right)^{{g}\left({x}\right)} \sim\left\{\left({x}−{a}\right){f}^{'} \left({a}\right)\right\}^{{g}\left({x}\right)} \:={e}^{{g}\left({x}\right){ln}\left\{\left({x}−{a}\right){f}^{'} \left({a}\right)\right\}} \\ $$$${if}\:{lim}_{{x}\rightarrow{a}} {g}\left({x}\right)=+\infty\:\:{we}\:{have}\:{lim}_{{x}\rightarrow{a}^{+} } {ln}\left\{\left({x}−{a}\right){f}^{'} \left({a}\right)\right\}=−\infty\:\:\Rightarrow \\ $$$${lim}_{{x}\rightarrow{a}^{+} } \:\:{g}\left({x}\right){ln}\left\{\left({x}−{a}\right){f}^{'} \left({a}\right)\right\}=−\infty\:\Rightarrow \\ $$$${lim}_{{x}\rightarrow{a}^{+} } \:\:\:\left\{{f}\left({x}\right)\right\}^{{g}\left({x}\right)} =\mathrm{0} \\ $$$${if}\:{lim}_{{x}\rightarrow{a}^{+} } \:\:{g}\left({x}\right)=−\infty\:\:\:\Rightarrow{lim}_{{x}\rightarrow{a}^{+} } \:\:{g}\left({x}\right){ln}\left\{\left({x}−{a}\right){f}^{'} \left({a}\right)\right\}=+\infty\:\Rightarrow \\ $$$${lim}_{{x}\rightarrow{a}^{+} } \:\:\:\left\{{f}\left({x}\right)\right\}^{{g}\left({x}\right)} \:=+\infty\:\:{and}\:{if}\:{you}\:{have}\:{a}\:{clear}\:{answer}\:{post}\:{it}.. \\ $$
Commented by Tony Lin last updated on 05/Nov/19
$${thanks}\:{sir} \\ $$
Commented by mathmax by abdo last updated on 05/Nov/19
$${you}\:{are}\:{welcome}. \\ $$
Answered by mind is power last updated on 04/Nov/19
$$\mathrm{f}\left(\mathrm{x}\right)^{\mathrm{g}\left(\mathrm{x}\right)} =\mathrm{e}^{\mathrm{g}\left(\mathrm{x}\right)\mathrm{ln}\left(\mathrm{f}\left(\mathrm{x}\right)\right)} \\ $$$$\mathrm{lim}\:\mathrm{g}\left(\mathrm{x}\right)=+\infty \\ $$$$\mathrm{lim}\left[\mathrm{f}\left(\mathrm{x}\right)\rightarrow\mathrm{0}\right. \\ $$$$\Rightarrow\mathrm{lim}\:\mathrm{ln}\left(\mathrm{f}\left(\mathrm{x}\right)\right)\rightarrow−\infty \\ $$$$\Rightarrow\mathrm{g}\left(\mathrm{x}\right)\mathrm{ln}\left(\mathrm{f}\right)\rightarrow−\infty \\ $$$$\Rightarrow\mathrm{e}^{\mathrm{g}\left(\mathrm{x}\right)\mathrm{ln}\left(\mathrm{f}\left(\mathrm{x}\right)\right)} \rightarrow\mathrm{0} \\ $$
Commented by Tony Lin last updated on 05/Nov/19
$${thanks}\:{sir} \\ $$
Answered by Tony Lin last updated on 05/Nov/19
$$\underset{{x}\rightarrow{a}} {\mathrm{lim}}{f}\left({x}\right)^{{g}\left({x}\right)} =\mathrm{0} \\ $$