Brief Idea of L'Hospital's Rule

Brief idea of L'Hospital's Rule. Consider two functions $f(x)$ and $g(x)$. If $\displaystyle\lim_{x\to a}f(x)=0$ and $\displaystyle\lim_{x\to a}g(x)=0$ (a $\frac00$ indeterminate form), and if $\displaystyle\lim_{x\to a}f'(x)=p$ and $\displaystyle\lim_{x\to a}g'(x)=q$ where $q\ne0$, then

$\lim_{x\to a}\dfrac{f(x)}{g(x)}=\lim_{x\to a}\dfrac{f'(x)}{g'(x)}=\dfrac pq$

If instead $\displaystyle\lim_{x\to a}g'(x)=0$ too, then — provided $\displaystyle\lim_{x\to a}f'(x)=0$ as well — the same $\frac00$ situation reappears one derivative down, and $\displaystyle\lim_{x\to a}\dfrac{f'(x)}{g'(x)}$ can itself be studied by the same rule (differentiating top and bottom again). …