Limit with $\tan$ L'Hopital

$ \infty - \infty $ does not a zero make, when it comes to limits.

For example, the limit \begin{equation*} \lim_{x \rightarrow \infty} \frac{x^5 - x^4}{x} \end{equation*} is clearly $ \infty $, but if you tried to "split" you would get $ \infty - \infty $. The problem is that the two quantities go to $ \infty $ at a different rate (here, the positive term is much faster than the negative term).

$$\lim_{x\to 0} \frac {\tan x}{x^3} -\frac{x}{x^3}=\lim_{x\to 0} \frac {\tan x}{x}\cdot\frac{1}{x^2} -\frac{1}{x^2}$$

and once,

$$\lim_{x\to 0} \frac {\tan x}{x}=1$$

your limit is $$1\cdot\infty - \infty=\infty - \infty$$ which is not $0$.