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

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

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

$$\underset{x\to 0}{\lim }(\frac{1}{x^2}-\frac{1}{\tan {}^{2}x})=$$$$\underset{x\to 0}{\lim }\left(\frac{(\tan x-x)(\tan x+x)}{x^2\tan {}^{2}x}\right)=$$$$\underset{x\to 0}{\lim }\left(\frac{\tan x+x}{x}\right)\times \underset{x\to 0}{\lim }\left(\frac{\tan x-x}{x{\tan }^{2}x}\right)=$$$$2\times \underset{x\to 0}{\lim }\frac{\frac{1}{2}\sin 2x-x{\cos }^{2}x}{x{\sin }^{2}x}=$$$$2\times \underset{x\to 0}{\lim }\frac{\cos 2x-[\cos {}^{2}x-x\sin 2x]}{\sin {}^{2}x+x\sin 2x}$$$$2\times \underset{x\to 0}{\lim }\frac{\cos {}^{2}x-1+x\sin 2x}{\sin {}^{2}x+x\sin 2x}$$$$2\times \underset{x\to 0}{\lim }\frac{-x^2+2x^2}{3x^2}=\frac{2}{3}$$$$$$

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