Question Number 213821 by hardmath last updated on 17/Nov/24
$$\mathrm{Find}:\:\:\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{sinx}}{\mathrm{x}}\right)^{\frac{\mathrm{sinx}}{\mathrm{x}\:−\:\mathrm{sinx}}} \:\:=\:\:? \\ $$
Answered by mehdee7396 last updated on 17/Nov/24
$${lim}_{{x}\rightarrow\mathrm{0}} \left(\frac{{sinx}}{{x}}−\mathrm{1}\right)\frac{{sinx}}{{x}−{sinx}} \\ $$$$={lim}_{{x}\rightarrow\mathrm{0}} \left(\frac{{sinx}−{x}}{{x}}\right)\frac{{sinx}}{{x}−{sinx}}=−\mathrm{1} \\ $$$$\Rightarrow{answer}={e}^{−\mathrm{1}} \: \\ $$$$ \\ $$