What are "instantaneous" rates of change, really?

Here's how I see it (please read the following if you can, because I address a lot of arguments people have already made):

Let's take instantaneous speed, for example. If it's truly instantaneous, then there is no change in $x$ (time), since there's no time interval.

Thus, in $\frac{f(x+h) - f(x)}{h}$, $h$ should actually be zero (not arbitrarily close to zero, since that would still be an interval) and therefore instantaneous speed is undefined.

If "instantaneous" is just a figure of speech for "very very very small", then I have two problems with it:

Firstly, well it's not instantaneous at all in the sense of "at a single moment".

Secondly, how is "very very very small" conceptually different from "small"? What's really the difference between considering $1$ second and $10^{-200}$ of a second?

I've heard some people talk about "infinitely small" quantities. This doesn't make any sense to me. In this case, what's the process by which a number goes from "not infinitely small" to "ok, now you're infinitely small"? Where's the dividing line in degree of smallness beyond which a number is infinitely small?

I understand $\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ as the limit of an infinite sequence of ratios, I have no problem with that.

But I thought the point of a limit and infinity in general, is that you never get there. For example, when people say "the sum of an infinite geometric series", they really mean "the limit", since you can't possibly add infinitely many terms in the arithmetic sense of the word.

So again in this case, since you never get to the limit, $h$ is always some interval, and therefore the rate is not "instantaneous". Same problem with integrals actually; how do you add up infinitely many terms? Saying you can add up an infinity or terms implies that infinity is a fixed number.

In math, there's intuition and there's rigor. Saying $$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}h$$ is a rigorous statement. It's very formal. Saying "the derivative is the instantaneous rate of change" is intuitive. It has no formal meaning whatsovever. Many people find it helpful for informing their gut feelings about derivatives.

(Edit I should not understate the importance of gut feelings. You'll need to trust your gut if you ever want to prove hard things.)

That being said, here's no reason why you should find it helpful. If it's too fluffy to be useful for you that's fine. But you'll need some intuition on what derivatives are supposed to be describing. I like to think of it as "if I squinted my eyes so hard that $f$ became linear near some point, then $f$ would look like $f'$ near that point." Find something that works for you.

The idea behind $$\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ is the slope of the graph $y=f(x)$ .

Now for a moment forget about your instantaneous velocity and think about your average velocity. What is average velocity? I think average velocity is $$\frac{x_f-x_i}{t_f-t_i}$$. Now if look at this carefully this is my slope but in a given interval of time.

So you might question what is the difference between instantaneous velocity and average velocity , Both of them talk about intervals .

No that not the idea over here. Now if it had been a linear graph it would have been very easy to calculate your instantaneous velocity , but in your you graph that's not the case. Here you see the particle (or object) is changing its velocity every moment of time and that becomes impossible to deal with . So to remove this element of doubt what we do is try to take the limit and try to get close to the frame of time and try finding the velocity and name it instantaneous velocity.