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Question Number 34522 by rahul 19 last updated on 07/May/18

lim_(x→0) log _e {((sin (a+(1/x)))/(sin a))}^x , 0<a<(π/2) .

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{log}\:_{{e}} \left\{\frac{\mathrm{sin}\:\left({a}+\frac{\mathrm{1}}{{x}}\right)}{\mathrm{sin}\:{a}}\right\}^{{x}} ,\:\mathrm{0}<{a}<\frac{\pi}{\mathrm{2}}\:. \\ $$

Commented byrahul 19 last updated on 08/May/18

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$$? \\ $$

Commented bymath khazana by abdo last updated on 09/May/18

let put A(x)=ln{((sin(a+(1/x)))/(sina))}^x we have A(x) =x { ln(a +(1/x)) −ln(sina)} =x{ ln(1+ax) −lnx −ln(sina)} =x ln(1+ax) −xlnx −x ln(sina) ⇒ lim_(x→0) A(x) =lim_(x→0) x ln(1+ax) but ln(1+ax) ∼ ax ⇒ lim_(x→0) A(x) =lim_(x→0) ax^2 =0

$${let}\:{put}\:{A}\left({x}\right)={ln}\left\{\frac{{sin}\left({a}+\frac{\mathrm{1}}{{x}}\right)}{{sina}}\right\}^{{x}} \:{we}\:{have} \\ $$ $${A}\left({x}\right)\:={x}\:\left\{\:{ln}\left({a}\:+\frac{\mathrm{1}}{{x}}\right)\:−{ln}\left({sina}\right)\right\} \\ $$ $$={x}\left\{\:{ln}\left(\mathrm{1}+{ax}\right)\:−{lnx}\:\:−{ln}\left({sina}\right)\right\} \\ $$ $$={x}\:{ln}\left(\mathrm{1}+{ax}\right)\:\:−{xlnx}\:\:−{x}\:{ln}\left({sina}\right)\:\Rightarrow \\ $$ $${lim}_{{x}\rightarrow\mathrm{0}} \:{A}\left({x}\right)\:={lim}_{{x}\rightarrow\mathrm{0}} \:{x}\:{ln}\left(\mathrm{1}+{ax}\right)\:\:\:{but} \\ $$ $${ln}\left(\mathrm{1}+{ax}\right)\:\sim\:{ax}\:\Rightarrow\:{lim}_{{x}\rightarrow\mathrm{0}} {A}\left({x}\right)\:\:\:={lim}_{{x}\rightarrow\mathrm{0}} \:{ax}^{\mathrm{2}} \:=\mathrm{0} \\ $$ $$ \\ $$

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