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lim-x-0-log-e-sin-a-1-x-sin-a-x-0-lt-a-lt-pi-2-




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 by rahul 19 last updated on 08/May/18
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$$? \\ $$
Commented by math 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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