A series and a ratio of derivatives

You will write three functions that test a limit by a series and by L'Hôpital's rule. The checker uses $\dfrac{x - \sin x}{x^3}$ and $\dfrac{e^x - 1 - x}{x^2}$.

The idea. The sine series is $x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dots$. Three terms are enough to see that $\dfrac{x - \sin x}{x^3}$ is close to $\dfrac{1}{6}$ for small $x$. L'Hôpital's rule replaces $\dfrac{f}{g}$ by $\dfrac{f'}{g'}$, so you can try the ratio of derivatives near the point.

Your tasks.

1. Finish sin_series(x, terms). Return the sum of the first terms terms of the sine series.
2. Finish ratio_by_series(x). Return (x - sin_series(x, 3)) / x ** 3.
3. Finish lhopital_ratio(fp, gp, x). Return fp(x) / gp(x), the ratio of the two derivatives given as functions.