L'Hopitals Rule

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!

$\displaystyle \displaystyle \frac{f(a + h) - 2f(a) + f(a-h)}{h^2} \to \frac{0}{0}$ as $\displaystyle \displaystyle h \to 0$. So you can use L'Hospital's Rule here...

Quote:

---

Originally Posted by worc3247

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

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!

---

The use of l'Hopital rule in the basic definition of second order derivative is not 'fully secure'... better is to derive the second derivative as limit as follows...

$\displaystyle \displaystyle f^{''} (a)= \lim_{h \rightarrow 0} \frac{\frac{f(a+h)-f(a)}{h} - \frac{f(a)-f(a-h)}{h}}{h}= \lim_{h \rightarrow 0} \frac{f(a+h)-2 f(a) + f(a-h)}{h^{2}}$

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$