I am solving a question which involves finding the continuity of a piece-wise function at a particular point, and it involves finding the below limit:$$\lim_{{x \to 0}} \frac{{\tan x - \sin x}}{{x^3}}$$
Now, my solving was to write $\frac{\sin x}{\cos x}$ instead of $\tan x$ and I finally got on to: $$\lim_{{x\to 0}}\frac{\sin x(1-\cos x)}{x^3\cos x}$$
Then I split it further as: $$\lim_{{x\to 0}}\frac{\sin x}{x}\times \lim_{{x \to 0}}\frac{(1-\cos x)}{x^2}\times\lim_{{x \to 0}}(\sec x)$$
The central term is bothering me.
L' Hopital would work fine, but I don't learn L' Hopital in school - it's only for the entrance examinations. Expansions are also not taught.
How do I proceed now?
