How to calculate the following limits?

Question

How is it possible calculate the following limits?

$\displaystyle \lim_{x\to 0} \frac{\arctan x -x}{x^3}$

$\displaystyle \lim_{x\to 0} \frac{\ln(1+x)-x}{x^2}$

$\displaystyle \lim_{x\to 0} \frac{\sinh x -x}{x^3}$

$\displaystyle \lim_{x\to 0} \frac{\cosh x -1}{x^2}$

I have calculated them using L'Hospital's rule.

For example:

$\displaystyle \lim_{x\to 0} \frac{\cosh x -1}{x^2}=$

$\displaystyle =\lim_{x\to 0} \frac{\sinh x}{2x}=$

$\displaystyle =\lim_{x\to 0} \frac{\cosh x}{2}=$

$=\displaystyle \frac{1}{2}$

But how is it possible to calculate them without using L'Hospital's rule nor Maclaurin series?

Answer 1

Hint : Try using Taylor series expansion to appropriate order in $x_{0} = 0$, aka Maclaurin series.