---
# System prepended metadata

title: Extension of L'hôpital's Rule
tags: [Calculus]

---

# Extension of L'hôpital's Rule
###### tags: `Calculus`
## Estimate Derivative
$$
f'(x)=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}h\approx\frac{f(x+h)-f(x)}h\\if\;f''(x)\;is\;continuous\\f''(x)=\lim_{h\rightarrow0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2}
$$
#### <Proof>
$$
\lim_{h\rightarrow0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2}\overset{L'H}=\lim_{h\rightarrow0}\frac{f'(x+h)-f'(x-h)}{2h}\\\overset{L'H}=\lim_{h\rightarrow0}\frac{f''(x+h)+f''(x-h)}2=f''(x)
$$
