How to solve $\lim\limits_{x\to 0} \frac{x – \sin(x)}{x^2}$ Without L'Hospital's Rule?
you can use trigonometric identities and inequalities, but you can't use series or more advanced stuff.
Calculus – Solving $\lim\limits_{x\to0} \frac{x – \sin(x)}{x^2}$ Without L’Hospital’s Rule
calculuslimitslimits-without-lhopital
Best Answer
The given expression is odd; therefore it is enough to consider $x>0$. We then have $$0<{x-\sin x\over x^2}<{\tan x -\sin x\over x^2}=\tan x\ {1-\cos x\over x^2}={\tan x\over2}\ \Bigl({\sin(x/2)\over x/2}\Bigr)^2\ ,$$ and right side obviously converges to $0$ when $x\to0+$.