TITLE: What are "instantaneous" rates of change, really?
QUESTION [50 upvotes]: 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.

REPLY [2 votes]: I had come to accept the slope intuition of derivatives a long time ago, when I first learned calculus. I never really understood it completely, I must confess, but since it proved useful in so many different applications (particularly in physics), I began to accept that interpretation without having to think about it at all after a certain point.
It was when I studied calculus slightly more formally that I gained a better understanding of the concept of limit and derivatives as a whole. The breakthrough came when I understood how sequences/series of numbers and their limit points work. Consider the following infinite sum:
\begin{equation}
1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...
\end{equation}
The 'limit' of this sum is equal to $2$. Now obviously, you can't go on adding infinite terms to get exactly $2$, yet we say (and we say this as a proper mathematical statement - not as an ambiguous expression of out intuition) that the limit of the sequence is $2$.
The idea behind the limit (in the case of sequences/series) is that if you kept on evaluating the sequence with more and more terms, you would come closer and closer to $2$. You wouldn't be 'converging' to $\textbf{any other number}$ other than $2$. We call $2$ a limit point of the sequence.
In a similar manner, when we talk about 'instantaneous' rates, what we are saying is that if we calculate the rate with smaller and smaller intervals, we get closer and closer to the value we call the 'instantaneous' rate. We never actually attain that value, but we get closer and closer to it.
In a sense, it's kind of like a 'sacred' value that your expression is always striving to reach. You can't ever really be sure of it even if you study the behaviour of the expression for very long. As an example, you might conclude that the sequence I showed above has a limit of $2.000000000001$, just by computing the value with more and more terms. But if you went even further, you would discover that the sum's distance from $2.000000000001$ begins to increase after some time (suppose you hit the midpoint of $2.000000000001$  and $2$, the true limit; continuing with the evaluation of the sum from there on out would lead you closer to $2$, and thus farther away from $2.000000000001$).
So you see, the thing about the limiting value of an expression is that it is not an easy task to determine it just by studying the way the expression behaves numerically (which is what the intuitive explanation of 'instantaneous' may lead you to believe). This is why the mathematics of limits is so profound. You can compute the $\textbf{exact}$ value that the expression converges to, regardless of 'practical artefacts' like numeric precision. It is actually quite an elusive value, the limit of an expression...but it has a lot of meaning to it. 
If you continued forever in the direction of the limit, you would continuously get closer to the limiting value (limit point), but you would never reach it. If we could measure the rate in an intervals of ever decreasing length, we would approach the instantaneous value - but we never get an exact match.