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


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Question Number 34522 by rahul 19 last updated on 07/May/18
$lim_(x→0) log _e {((sin (a+(1/x)))/(sin a))}^x , 0<a<(π/2) .$
$\lim_{x \to 0} \log_{e} {\frac{\sin (a + \frac{1}{x})}{\sin a}}^{x}, 0 < a < \frac{π}{2} .$
Commented byrahul 19 last updated on 08/May/18

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Commented bymath khazana by abdo last updated on 09/May/18
$let put A(x)=ln{((sin(a+(1/x)))/(sina))}^x we have A(x) =x { ln(a +(1/x)) −ln(sina)} =x{ ln(1+ax) −lnx −ln(sina)} =x ln(1+ax) −xlnx −x ln(sina) ⇒ lim_(x→0) A(x) =lim_(x→0) x ln(1+ax) but ln(1+ax) ∼ ax ⇒ lim_(x→0) A(x) =lim_(x→0) ax^2 =0$
$l e t p u t A (x) = l n {\frac{s i n (a + \frac{1}{x})}{s i n a}}^{x} w e h a v e$ $A (x) = x {l n (a + \frac{1}{x}) - l n (s i n a)}$ $= x {l n (1 + a x) - l n x - l n (s i n a)}$ $= x l n (1 + a x) - x l n x - x l n (s i n a) \Rightarrow$ ${l i m}_{x \to 0} A (x) = {l i m}_{x \to 0} x l n (1 + a x) b u t$ $l n (1 + a x) \sim a x \Rightarrow {l i m}_{x \to 0} A (x) = {l i m}_{x \to 0} {a x}^{2} = 0$
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