find-lim-x-0-sinx-x-1-1-cosx-

Question Number 29459 by prof Abdo imad last updated on 08/Feb/18

f i n d {l i m}_{x \to 0} {(\frac{s i n x}{x})}^{\frac{1}{1 - c o s x}} .

Commented by Cheyboy last updated on 09/Feb/18

\frac{1}{\sqrt[3]{e}}

Commented by mrW2 last updated on 09/Feb/18

t o c h e y b o y s i r :

s i r, i t w o u l d m a k e m o r e s e n s e, i f y o u

c o u l d a l s o s h o w y o u r w o r k i n g i n s t e a d

o f o n l y a r e s u l t . t h a n k y o u!

Commented by Cheyboy last updated on 09/Feb/18

s i r i w a s h a v i n g l o w b a t t e r y t h a t s

w h y i c o u l d n o t t y p e t h e w o r k i n g .

t h a n k s f o r t h a t s i r i w i l l s t i c k t o

y o u r w o r d n e x t t i m e .

Commented by mrW2 last updated on 09/Feb/18

t h a n k s!

Commented by prof Abdo imad last updated on 10/Feb/18

l e t p u t A (x) = {(\frac{s i n x}{x})}^{\frac{1}{1 - c o s x}} \Rightarrow l n (A (x)) = \frac{1}{1 - c o s x} l n (\frac{s i n x}{x})

s i n x = x - \frac{x^{3}}{3!} + o (x^{5}) \Rightarrow \frac{s i n x}{x} = 1 - \frac{x^{2}}{7} + o (x^{4}) a n d

l n (\frac{s i n x}{x}) = l n (1 - \frac{x^{2}}{6} + o (x^{4})) \sim - \frac{x^{2}}{6} w e h a v e a l s o

c o s x \sim 1 - \frac{x^{2}}{2} \Rightarrow 1 - c o s x \sim \frac{x^{2}}{2} s o

\frac{1}{1 - c o s x} l n (\frac{s i n x}{x}) \sim \frac{2}{x^{2}} \times (- \frac{x^{2}}{6}) = - \frac{1}{3} \Rightarrow

{l i m}_{x \to 0} l n (A (x)) = - \frac{1}{3} \Rightarrow {l i m}_{x \to 0} A (x) = e^{- \frac{1}{3}}

= {(^{3} \sqrt{e})}^{- 1} .

Commented by ajfour last updated on 10/Feb/18

t h a n k s f o r s o l u t i o n, S i r .

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