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

Question Number 42786 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{1 - \frac{x}{s i n x}}{x^{2}}

Commented by maxmathsup by imad last updated on 30/Sep/18

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

\frac{s i n x - x}{x^{2} s i n x} \sim \frac{- \frac{x^{3}}{3!}}{x^{3}} \to - \frac{1}{6} w h e n x \to 0 \Rightarrow

{l i m}_{x \to 0} \frac{1 - \frac{x}{s i n x}}{x^{2}} = - \frac{1}{6} .

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

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

\lim_{x \to 0} \frac{c o s x - 1}{x^{2} c o s x + s i n x . 2 x} (\frac{0}{0})

\lim_{x \to 0} \frac{s i n x}{- x^{2} s i n x + c o s x . 2 x + s i n x . 2 + 2 x c o s x} (\frac{0}{0})

\lim_{x \to 0} \frac{\frac{s i n x}{x}}{- x s i n x + 2 c o s x + 2 \frac{s i n x}{x} + 2 c o s x}

= \frac{1}{- 0 + 2 + 2 + 2} = \frac{1}{6}