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 xx (time), since there's no time interval.
Thus, in 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 limh→0f(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.
