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

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).

Example 1: $\displaystyle\lim_{x\to0}\dfrac{\sin x}{x^2}$. Here $f(x)=\sin x$ with $\lim_{x\to0}f(x)=0$, and $g(x)=x^2$ with $\lim_{x\to0}g(x)=0$. Then $f'(x)=\cos x$, so $\lim_{x\to0}f'(x)=\cos0=1\ne0$; but $g'(x)=2x$, so $\lim_{x\to0}g'(x)=2(0)=0$. Since $\lim_{x\to0}g'(x)=0$ (and $f'$'s limit is not also $0$), L'Hospital's Rule cannot be applied here. …