Differantiation proof question

[transgalactic]

if f(x) is differentiable on x_0
prove that
http://img102.imageshack.us/img102/3189/51290270sj1.th.gif

??

using the given i could
say that lim [f(x+x_0) - f(x0)]/[x-x_0] exist

 

[quasar987]

You sure could. But a little more to the point, you could say that

$$f'(x_0)=\lim_{h\rightarrow 0}\frac{f(x_0+h)-f(x_0)}{h}=\lim_{h\rightarrow 0}\frac{f(x_0-h)-f(x_0)}{-h}$$

The rest is a judicious use of algebra.

[Aren't you missing the limit sign and the "0" indices attached to the x's in you image?]