find-lim-x-0-ln-x-e-sinx-x-2-sh-2x-

Question Number 37816 by prof Abdo imad last updated on 17/Jun/18

f i n d {l i m}_{x \to 0} \frac{l n (x + e^{s i n x}) - x^{2}}{s h (2 x)}

Commented by math khazana by abdo last updated on 28/Jun/18

l e t u s e h o s p i t a l t h e o r e m l e t f (x) = l n (x + e^{s i n x}) - x^{2}

a n d g (x) = s h (2 x) \Rightarrow

f^{^{'}} (x) = \frac{1 + {c o s x e}^{s i n x}}{x + e^{s i n x}} - 2 x \Rightarrow {l i m}_{x \to 0} f (x) = 2

g^{^{'}} (x) = 2 c h (2 x) \Rightarrow {l i m}_{x \to 0} g^{^{'}} (x) = 2 \Rightarrow

{l i m}_{x \to 0} \frac{l n (x + e^{s i n x}) - x^{2}}{s h (2 x)} = 1