洛必达法则的直觉

L'Hospital's Rule is actually a very intuitive theorem in calculus. Let us examine the two indeterminate cases and see how l'Hospital's Rule can be easily seen.

For example, suppose that we have a $\frac{0}{0}$ indeterminate limit at a value $a.$ In this case, we are interested in $\lim_{x\rightarrow a}\frac{f(x)}{g(x)},$ where $f$ and $g$ are continuous, differentiable functions that approach $0$ as $x$ approaches $a.$

Then, very close to $a,$ we have: $$f(x)\approx f'(x)(x-a)$$ $$g(x)\approx g'(x)(x-a)$$ $$\frac{f(x)}{g(x)}\approx\frac{f'(x)}{g'(x)}.$$ This intuitively demonstrates l'Hospital's Rule for the $\frac{0}{0}$ indeterminate limit.