- #1

Unassuming

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## Homework Statement

part 1. )

If f:(a,b)--> R is differentiable at p in (a,b), prove that

f ' (p) = lim (n -> oo) n [ f(p + 1/n) - f(p) ].

part 2. )

Show by example that the existence of the limit of the sequence,

{n [ f(p + 1/n) - f(p) ] } does not imply the existence of f ' (p).

## Homework Equations

## The Attempt at a Solution

One failed attempt involved comparing this limit to the definition below,

[tex] \lim_{n \rightarrow \infty} \frac{f(p_n)-f(p)}{p_n-p} [/tex].

I attempted to break the new definition (up top) into the sum of the limits and proceed down that path but it did not lead to anything.

I have also tried this new definition with a functions and a point and compared its result to f'(p) to gain some understanding. I still cannot seem to get anywhere though.

Any help would be appreciated. Thank you.