Definition of derivative - is it approaching value or actual value?

Question

When we say a limit, say for example: $\lim_\limits{x\to 2} f(x)$, where $f(x) = x^2$.

And we say this that as $x$ tends to $2$, the value of $f(x)$ approaches to $4$. (I.e., the actual value of the function as $x$ tends to $2$ might not be $4$, but it surely is approaching to $4$ from either side.)

Then, we used this definition of limit to define derivative of a function at a point:

$$\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$

But while saying the value of derivative at some $x$, we say absolutely the value, not approaching.

For example, we say the derivative of $x^2$ at $x=3$ is $6$. Shouldn't it be said that the derivative of $x^2$ at $x=3$ approaches $6$?

Answer 1

"Approaches" is more of colloquial term. There is a rigorous definition of the limit, and it says that the limit is (exactly!) equal to something (If the limit exists).

The derivative of $x^2$ at $x=3$ is equal to $6$, not approaching $6$.

Answer 2

Let $f(x)=x+1$. So $\lim_{x\to 2}f(x)=3$. That's the same as saying "the limit of $f(x)$ as $x$ approaches $2$ is $3$". It's also the same as saying "$f(x)$ approaches $3$ as $x$ approaches $2$".

But not "the limit of $f(x)$ approaches $3$"!!! Students say that sometimes - they shouldn't. That limit is a number. One single number, doesn't "approach" anything, in fact it equals $3$.