find-lim-x-0-sin-shx-sh-sinx-x-

Question Number 35727 by abdo mathsup 649 cc last updated on 22/May/18

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

Commented by abdo mathsup 649 cc last updated on 23/May/18

i t s b e t e r t o u s e h o s p i t a l t h e o r e m i n t h i s c a s e

l e t p u t f (x) = s i n (s h x) - s h (s i n x) a n d

g (x) = x w e h a v e

f^{^{'}} (x) = c h (x) c o s (s h x) - c o s (x) c h (s i n x) \Rightarrow

{l i m}_{x \to 0} f (x) = 1 - 1 = 0

g^{^{'}} (x) = 1 s o {l i m}_{x \to 0} \frac{s i n (s h x) - s h (s i n x)}{x} = 0