# 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.

• See also my comment following the answer of Alex G. Commented Oct 30, 2021 at 5:37

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.

• +1 : nicely phrased. You might also mention that one of the consequences of the definition of the derivative of $f(x)$ at the point $(x = x_0)$ is that for the derivative of $f(x)$ to exist at $(x = x_0)$, the function $f(x)$ must be continuous at $(x = x_0)$. This might not be readily apparent to the OP (i.e. original poster). That is, in general, for a typical function $h(x)$, you can have that the $\displaystyle \lim_{x \to x_0} h(x)$ exists and equals some finite value $L$, without the function $h(x)$ actually being continuous at $(x = x_0)$. Commented Oct 30, 2021 at 5:35
• This completely clears up my misunderstanding. Thanks! Commented Oct 30, 2021 at 17:28

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.