i) Let $\displaystyle f:\mathbb{R}\rightarrow\mathbb{R}$ be differentiable, let $\displaystyle a\in\mathbb{R}$. Suppose that $\displaystyle f$''$(a)$ exists.

Prove that $\displaystyle \displaystyle\lim_{h\to 0}\frac{f(a+h)-2f(a)+f(a-h)}{h^2} = f$''$(a)$.

ii) Suppose further that $\displaystyle f$''$(a) $ exists for all x and that $\displaystyle f$'''$(0) $ exists. Prove that $\displaystyle \displaystyle\lim_{h\to 0}\frac{4(f(h)-f(-h)-2(f(\frac{h}{2})-f(-\frac{h}{2})))}{h^3}=f$'''$(0)$.

I realise I need to use L'Hopitals rule at some point but i'm not sure how I can justify the conditions needed to use it. If someone could help me with the first part, I may be able to get the second. Thanks!