Proof identity of differential equation I would appreciate if somebody could help me with the following problem:
Q: $f''(x)$ continuous in $\mathbb{R}$ show that 
$$ \lim_{h\to 0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2}=f''(x)$$ 

Use  L'Hospital's rule
$$ \lim_{h\to 0}\frac{F(h)}{G(h)}=\lim_{h\to 0}\frac{F'(h)}{G'(h)}$$
You can use the rule if you have $\frac{0}{0}$ result.
You need to apply it 2 times
$$ \lim_{h\to 0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2}= \lim_{h\to 0}\frac{f'(x+h)-f'(x-h)}{2h}=\lim_{h\to 0}\frac{f''(x+h)+f''(x-h)}{2}=f''(x)$$ 
