lim-x-0-1-x-2-cot-2-x-

Question Number 83859 by john santu last updated on 06/Mar/20

\lim_{x \to 0} (\frac{1}{x^{2}} - \cot^{2} x) = ?

Commented by john santu last updated on 06/Mar/20

\lim_{x \to 0} (\frac{1}{x^{2}} - \frac{1}{\tan^{2} x}) =

\lim_{x \to 0} (\frac{(\tan x - x) (\tan x + x)}{x^{2} \tan^{2} x}) =

\lim_{x \to 0} (\frac{\tan x + x}{x}) \times \lim_{x \to 0} (\frac{\tan x - x}{x \tan^{2} x}) =

2 \times \lim_{x \to 0} \frac{\frac{1}{2} \sin 2 x - x \cos^{2} x}{x \sin^{2} x} =

2 \times \lim_{x \to 0} \frac{\cos 2 x - [\cos^{2} x - x \sin 2 x]}{\sin^{2} x + x \sin 2 x}

2 \times \lim_{x \to 0} \frac{\cos^{2} x - 1 + x \sin 2 x}{\sin^{2} x + x \sin 2 x}

2 \times \lim_{x \to 0} \frac{- x^{2} + 2 x^{2}}{3 x^{2}} = \frac{2}{3}