Question about defn. of derivative

[AxiomOfChoice]

Let \{h_n\} be ANY sequence of real numbers such that h_n \neq 0[/tex] and [itex]h_n \to 0. If f'(x) exists, do we have


f'(x) = \lim_{n\to \infty} f_n(x),


where

f_n(x) = \frac{1}{h_n} (f(x+h_n) - f(x))


????

This seems to express the derivative as the pointwise limit of a sequence of functions...right? Do we know, in addition, that the convergence is uniform?

[mathman]

Uniform in what sense? The definition does not require uniformity in x.

[AxiomOfChoice]

Uniform in what sense? The definition does not require uniformity in x.

Ok, I'm not sure :) I didn't really think before writing out that question. But am I right on all other counts?

[mathman]

You are right, but it is more cumbersome than the usual approach where:

f'(x)=lim(h->0) (f(x+h) - f(x))/h

[HallsofIvy]

lim_{x\to a} f(x)= L if and only if \lim_{n\to \infty} f(a_n)= L for every sequence \{a_n\} that converges to a. The two formulations are equivalent.