Question Number 34522 by rahul 19 last updated on 07/May/18
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{log}\:_{{e}} \left\{\frac{\mathrm{sin}\:\left({a}+\frac{\mathrm{1}}{{x}}\right)}{\mathrm{sin}\:{a}}\right\}^{{x}} ,\:\mathrm{0}<{a}<\frac{\pi}{\mathrm{2}}\:. \\ $$
Commented by rahul 19 last updated on 08/May/18
$$? \\ $$
Commented by math khazana by abdo last updated on 09/May/18
$${let}\:{put}\:{A}\left({x}\right)={ln}\left\{\frac{{sin}\left({a}+\frac{\mathrm{1}}{{x}}\right)}{{sina}}\right\}^{{x}} \:{we}\:{have} \\ $$$${A}\left({x}\right)\:={x}\:\left\{\:{ln}\left({a}\:+\frac{\mathrm{1}}{{x}}\right)\:−{ln}\left({sina}\right)\right\} \\ $$$$={x}\left\{\:{ln}\left(\mathrm{1}+{ax}\right)\:−{lnx}\:\:−{ln}\left({sina}\right)\right\} \\ $$$$={x}\:{ln}\left(\mathrm{1}+{ax}\right)\:\:−{xlnx}\:\:−{x}\:{ln}\left({sina}\right)\:\Rightarrow \\ $$$${lim}_{{x}\rightarrow\mathrm{0}} \:{A}\left({x}\right)\:={lim}_{{x}\rightarrow\mathrm{0}} \:{x}\:{ln}\left(\mathrm{1}+{ax}\right)\:\:\:{but} \\ $$$${ln}\left(\mathrm{1}+{ax}\right)\:\sim\:{ax}\:\Rightarrow\:{lim}_{{x}\rightarrow\mathrm{0}} {A}\left({x}\right)\:\:\:={lim}_{{x}\rightarrow\mathrm{0}} \:{ax}^{\mathrm{2}} \:=\mathrm{0} \\ $$$$ \\ $$