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

Question Number 42788 by maxmathsup by imad last updated on 02/Sep/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 maxmathsup by imad last updated on 30/Sep/18

w e h a v e \frac{2 x}{l n (\frac{1 + x}{1 - x})} = \frac{2 x}{l n (1 + x) - l n (1 - x)} b u t

l n (1 + x) \sim x a n d l n (1 - x) \sim - x \Rightarrow l n (1 + x) - l n (1 - x) \sim 2 x \Rightarrow

{l i m}_{x \to 0} \frac{2 x}{l n (\frac{1 + x}{1 - x})} - c o s x = {l i m}_{x \to} \frac{2 x}{2 x} - 1 = 1 - 1 = 0 .

Answered by tanmay.chaudhury50@gmail.com last updated on 04/Sep/18

\lim_{x \to 0} \frac{2 x}{2 (x + \frac{x^{3}}{3} + \frac{x^{5}}{5} \dots)} - c o s x

\lim_{x \to 0} \frac{1}{1 + \frac{x^{2}}{3} + \frac{x^{4}}{5} + . .} - c o s x

= 1 - 1 = 0