Question Number 99935 by student work last updated on 24/Jun/20
$$\mathrm{li}\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{m}}\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{ln}\left(\mathrm{e}^{\mathrm{x}} −\mathrm{1}\right)} }=? \\ $$$$\mathrm{help}\:\mathrm{me} \\ $$
Commented by student work last updated on 24/Jun/20
$$\mathrm{help}\:\mathrm{me}\:\mathrm{sisrs} \\ $$
Commented by student work last updated on 24/Jun/20
$$ \\ $$
Commented by student work last updated on 24/Jun/20
$$\mathrm{who}\:\mathrm{can}\:\mathrm{help}\:\mathrm{me}? \\ $$
Commented by student work last updated on 24/Jun/20
$$\mathrm{l}\:\mathrm{need}\:\mathrm{your}\:\mathrm{help} \\ $$
Answered by bobhans last updated on 24/Jun/20
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{x}^{−\mathrm{ln}\left(\mathrm{e}^{\mathrm{x}} −\mathrm{1}\right)} \:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{1}}{\mathrm{x}}\right)^{\mathrm{ln}\left(\mathrm{e}^{\mathrm{x}} −\mathrm{1}\right)} \\ $$$$=\:\mathrm{e}^{\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{ln}\left(\mathrm{e}^{\mathrm{x}} −\mathrm{1}\right).\:\mathrm{ln}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)} \\ $$$$ \\ $$