Relation between "$\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$" and "$\lim_{x \rightarrow a} f(x)$"? 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.
 A: The definition of the derivative involves the limit of a new function, not your original function $f$. We fix a value $x$ at which we wish to compute the derivative of $f$. Then we define a new function (call it $g$ for lack of creativity):
$$
g(h) = \frac{f(x + h) - f(x)}{h}
$$
Now we note that $g$ is defined on a small neighborhood of $0$, and so we can consider the limit $\lim_{h \to 0} g(h)$. If this limit exists, it is, by definition, $f'(x)$.
People often leave the function $g(h)$ "anonymous" rather than giving it a name like $g(h)$, but it is understood from context that in the limit you're asking about, $x$ is fixed and $h$ is varying. This may take a bit of getting used to.
A: Intuitively, the limit of a function $f(x)$ at $a$ is just the value of $f$ near $a$. This is quite simple to visualize.
On the other hand, the (first) derivative of a function at $a$ is the "limit" of how fast the function changes near $a$. Graphically, it is the slope of the function.
More formally, the derivative $f'$ is the limit of the function that computes the slope of $f$. So technically it's a limit but it has a specific purpose, to compute a slope.
