Question

Is there a difference between `lim_(h->0)(f(x+h)-f(x))/h` and `lim_(deltax->0)(f(x+deltax)-f(x))/(deltax)`?

Answer 1

Yes, there is a difference since the first limit is defined at `x=0`, but the second one is not.

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I hope that this was helpful.

Answer 2

It is only a different notation.

Explanation:

I've never seen this with a lower case `delta`, but the principle is the same as for `Deltax`

We treat `Deltax` (or`deltax`) as a single variable written using two symbols. So we are just using `Deltax = delta x = h` .

You might say that the `Deltax` notation emphasizes that we are taking a change in `x` and asking about a limit as that change approaches `0`.
(The `h` is a bit mysterious to many students.)

Answer 3

The difference is only the notation. Personally, I find it clearer to use either `h` or `Deltax` because `Deltax` implies that for `x_2 - x_1`, the value of `x_1` doesn't matter, and so if `x_1 = 0`, then we can rename `x_2` to `h` and now the notation matches.

I typically see `h` associated with `x`, and `k` associated with `y`.

The point of `h` vs. `deltax` vs. `Deltax` is that it's just the length of the examination window. One could easily say `epsilon` to indicate this small length.

The whole idea of this limit derivative definition is to say that when you zoom into a graph (with your calculator, for instance), the more you zoom in, the more linear the graph looks (and the smaller this `Deltax` is). The more linear it looks, the better the conditions you have for saying "the slope is only the rise over the run". To convert the basic slope formula into the new notation, you note that the typical slope formula is:

`(y_2 - y_1)/(x_2 - x_1)`

and you modify it to use this new miniscule window width, `(x_2 - x_1) -> 0`, and the new miniscule window height, `f(x+[Deltax->0]) - f(x)`:

`(Deltaf(x))/(Deltax) => lim_(Deltax->0) (f(x+Deltax) - f(x))/(x) = lim_(Deltax->0) (f(x+h) - f(x))/(h)`

where `x_1 = 0`.

So really, this is saying that since this slope exactly lines up with the graph on a really close zoom (`Deltax = h` is small), the derivative is the slope at that point.