calculate-lim-x-0-2x-ln-1-x-1-x-cosx-

Question Number 40118 by maxmathsup by imad last updated on 15/Jul/18

c a l c u l a t e {l i m}_{x \to 0} \frac{2 x}{l n (\frac{1 + x}{1 - x})} - c o s x

Commented by math khazana by abdo last updated on 19/Jul/18

l e t A (x) = \frac{2 x}{l n (\frac{1 + x}{1 - x})} - c o s x

A (x) = \frac{2 x}{l n (1 + x) - l n (1 - x)} - c o s x b u t

{l n}^{^{'}} (1 + x) = \frac{1}{1 + x} = 1 - x + o (x^{2}) \Rightarrow

l n (1 + x) = x - \frac{x^{2}}{2} + o (x^{3}) a n d

l n (1 - x) = - x - \frac{x^{2}}{2} + o (x^{3}) \Rightarrow

l n (1 + x) - l n (1 - x) = 2 x + o (x^{3}) a l s o

c o s x \sim 1 - \frac{x^{2}}{2} (x \to 0) \Rightarrow

A (x) \sim 1 - (1 - \frac{x^{2}}{2}) \Rightarrow {l i m}_{x \to 0} A (x) = 0 .