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