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 \to 0} \log_{e} {\frac{\sin (a + \frac{1}{x})}{\sin a}}^{x}, 0 < a < \frac{π}{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

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