I have rewritten this entire question, since what I've learned since asking it requires me to restate it. I want to get rid of the obfuscating revisions.
Let's say that f is a continuous function.
$f(x)$ approaches L as x approaches a. So $\lim\limits_{x \to a}f(x) = L$
When it's said that the gradient of a tangent line to a curve at some particular point has some particular value, this is the same as saying f(a)=L. But without explicitly evaluating at that point, you can't say as much. All you can say is what happens as you approach that value. In other words, you can only say what value $f(x)$ approaches as $x$ approaches $a$, you can't say what $f(a)$ is.
Is that true?
Some textbooks will just say that the value of $f(a)=L$. Sal Khan's explanation does this. He says, about the function as it approaches the limit, "this is the gradient of the tangent." I'm saying that it should be said that, "the derivative approaches the gradient of the tangent to the curve." I think "approaches" and "is" are very different.
If there is a way to prove that the $f(a)=L$ then I'd like to see it. I don't know how to do this yet.
Best Answer
The confusion seems to be in your line
It is $f(x)$ that approaches $L$.
That is, the function $f(x)$ approaches some value as $x$ tends to $a$. That value is called the limit, and is denoted $\lim\limits_{x\to a}{f(x)}$. Saying $\lim\limits_{x\to a}{f(x)}=L$ is saying these two values are equal.
The derivative $\frac{dy}{dx}$ is the gradient of the tangent line, which is the limit of gradients of successively smaller secants. Again, a limit is a value, not something that approaches a value.
Edit: let me see whether I can help with your rewritten question.
Suppose $f$ is a continuous function. Then $\lim\limits_{x\to a}f(x) = f(a)$ (this is by definition of continuity -- we haven't taken derivatives yet).
Now suppose $f$ is differentiable at $a$, and suppose $f'(a) = L$ -- that is, $$\lim\limits_{h \to 0} \frac{f(a+h)-f(a)}{h} = L$$.
The function $$g(h) = \frac{f(a+h)-f(a)}{h}$$, which is defined for $h$ non-zero, represents the gradients of secants. The function $g$ approaches $L$ as $h$ tends to $0$ -- this is what it means for $f$ to be differentiable at $a$. So $g$ approaches the gradient of the tangent to the curve at $a$. But $g$ is not the derivative. The derivative is $\lim\limits_{h\to 0}g(h)$, which is $L$.