calculate-lim-x-0-cos-x-sinx-1-x-2-

Question Number 40709 by math khazana by abdo last updated on 26/Jul/18

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

Commented by maxmathsup by imad last updated on 26/Jul/18

w e h a v e s i n x = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n + 1)!} x^{2 n + 1} \Rightarrow s i n x = x - \frac{1}{3!} x^{3} + o (x^{5}) \Rightarrow

x - s i n x = \frac{x^{3}}{6} + o (x^{5}) \Rightarrow c o s (x - s i n x) = c o s (\frac{x^{3}}{6} + o (x^{5})) \sim 1 - \frac{x^{6}}{12} (x \to 0) \Rightarrow

c o s (x - s i n x) - 1 \sim - \frac{x^{6}}{12} \Rightarrow \frac{c o s (x - s i n x) - 1}{x^{2}} \sim \frac{x^{4}}{12} \Rightarrow

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

Answered by tanmay.chaudhury50@gmail.com last updated on 27/Jul/18

\lim_{x \to 0} \frac{- 2 {s i n}^{2} (\frac{x - s i n x}{2})}{{(\frac{x - s i n x}{2})}^{2}} \times \frac{{(\frac{x - s i n x}{2})}^{2}}{x^{2}}

w h e n x \to 0 \frac{x - s i n x}{2} \to 0

t = \frac{x - s i n x}{2}

\lim_{t \to 0} \frac{- 2 {s i n}^{2} t}{t^{2}} \times \lim_{x \to 0} \frac{x^{2} {(1 - \frac{s i n x}{x})}^{2}}{4 x^{2}}

= - 2 \times \frac{1}{4} \times (1 - 1) = 0