# Challenge. (1)

• Mar 6th 2010, 03:25 PM
Miss
Challenge. (1)
Hello :)

Problem:
Find a limit definition for $f''(x)$ similar for $f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$, Your limit should not have any $f'$. i.e., it must be in terms of $f$ only.
• Mar 6th 2010, 04:37 PM
TheEmptySet
Quote:

Originally Posted by Miss
Hello :)

Problem:
Find a limit definition for $f''(x)$ similar for $f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$, Your limit should not have any $f'$. i.e., it must be in terms of $f$ only.

define

$g(x)=\frac{f(x+h)-f(x)}{h}$

Then

$\lim_{h \to 0}\frac{g(x+h)-g(x)}{h}=\lim_{h \to 0}\frac{f(x+2h)-2f(x+h)+f(x)}{h^2}$