Evaluation of limit $\displaystyle \lim_{x\rightarrow 0}\frac{\cos (\tan x)-\cos x}{x^4}\;,$

without using L, Hospital Rule and series expansion.

I have solved it using $\displaystyle \cos (\tan x)-\cos x = -2\sin\left(\frac{\tan x+x}{2}\right)\cdot \sin \left(\frac{\tan x-x}{2}\right)$

Now Using $\displaystyle \lim_{x\rightarrow 0}\frac{\tan x-x}{x^3} = \frac{1}{3}$

But i did not understand how can i solve it without using $\displaystyle $ L hospital rule and series exp.

Help required, Thanks


$$=\dfrac{\sin\dfrac{\tan x+x}2}{\dfrac{\tan x+x}2}\cdot\dfrac{\dfrac{\tan x+x}2}{\dfrac x2} \cdot\dfrac{\sin\dfrac{\tan x-x}2}{\dfrac{\tan x-x}2}\cdot\dfrac{\tan x-x}{x^3}$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.