Perhaps I'm not understanding of it myself but to me there is a 'disconnect' between limit and derivative

> 
> definition of the limit:
> 
> 
> Let a function f(x) be defined on a deleted neighborhood around x = a.
> Then we say that, $\lim\_{x \rightarrow a} f(x) = L$
> 
> 
> if for every $\varepsilon > 0$, there is some $\delta > 0$ s.t. $|f(x) - L| < \varepsilon $ whenever $0 < |x-a| < \delta$
> 
> 
> definition of derivative:
> 
> 
> $$ f'(x) = \lim\_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$
> 
> 
> 

My problem/question is that the definition of limit is to a function and in a derivative which has a limit built-in into it, it is to a variable, h, not related to a function. Am I wrong in this thinking? (feel free to answer in terms of analysis if necessary)

From what I understand, a limit has you take closer and closer approximations of a function around a particular value. A derivative has you take slopes whose points get closer and closer hence why the h goes to 0.