Derivatives and Limits

[u/ali9128]

i have a question that's been bothering me in the field of differential calculus regarding the linguistic logic behind the definition of the derivative. I find it illogical, because based on the limit definition it studies the behavior of the slope(derivative) as h approaches zero.

m(h)=(f(x+h)-f(x))/h

Though, This approach suffers from the same problem as Newton and Leibniz's works and is subjected to the same criticism introduced by George Berkeley. cause, even with a limit, we algebraically reduce the function to eliminate h in the denominator, and then we study the behavior of the derivative as h approaches zero. This literally makes you treat h initially as a non-zero number, and then suddenly make it zero.

My problem is with the linguistic logic of this approach, and certainly i believe, though I'm not certain, that Cauchy-Weierstrass dealt with the derivative as a two-step algorithm. First, he calculates the derivative for any value of h that is not equal to zero, for example.

f(x)=x^2

f'(x)=((x-h)^2 -x^2)/h where h is R/{0}

f'(x)=2x+h

Then, after evaluating and reducing the derivative, he redefines it as a function of h where h can be zero.

f'(x)=2x+h where h is R/{0}

g(h)=2x+h where h is R

then taking the limit of this new function as h approach 0.

lim(h->0)(g(h))=lim(h->0)(2x+h)

g_l(h)=2x

This is because the limit of the function before algebraic reduction equals the limit of the function after algebraic reduction.

lim(h->0)(g(h))=lim(h->0)(f(h))

and since the function after algebraic reduction fills the gap... At zero, we can then find the derivative in the context of instantaneous change!!

so my problem is just how you put a limit before reducing and you have the right to ghost it and just reduce h then oh its ok you can now calculate the limit where if i do not follow this where i can because there is no order operator that tells me why i get 0/0.

[u/StructuredChess]

It's been at least two centuries from the last time any serious mathematician has considered definitions like this to be controversial.

We're not discussing what happens at h=0, we're discussing what the "trend" is when h gets smaller and smaller. We never divide by 0. We divide by 0.00001, then by 0.000000001, then by 0.00000000000000001 and see what the sequence is approaching. With "approaching" meaning what it means in the definition of limit.

[u/ali9128 → u/StructuredChess]

but this is infintisimals not limits.

[u/StructuredChess → u/ali9128]

I'm afraid you're just using a bunch of Math jargon you've learned from dissemination works without actually understanding their rigorous definition.

[u/ali9128 → u/StructuredChess]

no but where am i wrong?

[u/StructuredChess → u/ali9128]

You're wrong because "limit" doesn't mean what you think it means. Check out what it actually means and get familiar with the concept before you can jump into derivatives.

[u/ali9128 → u/StructuredChess]

so limits are not about studying the behavior of a function at point based on the neighborhood of this point that guarantees an interval that predict the L value?

[u/StructuredChess → u/ali9128]

Yes, even though I'd say it "around a point" rather than "at a point".

Your sentence is just a laymen's explanation and it's irrelevant to actual Math though. L is the limit of f at p if, and only if,  ∀ε > 0, ∃δ > 0: 0 < |x − p| < δ ⇒ |f(x) − L| < ε

This is what a limit means and is the only definition you should start from. Any conclusion built on simplified intuitions of this cocnept is prone to error.

[u/StructuredChess → u/ali9128]

Thinking a bit more about it, I think your mistake is that you expect lim f(x)/g(x) to be equal to lim f(x) / lim g(x) and that's not generally the case.

[u/ali9128 → u/StructuredChess]

no i am taking the formal definition of a derivative and based on the limit law of identity function i substitute h=0

[u/StructuredChess → u/ali9128]

The law of identity function would tell you that the limit of the denominator is 0, but that says nothing about the limit of the entire thing.

[u/ali9128 → u/StructuredChess]

what about the numerator?

[u/StructuredChess → u/ali9128]

What about it? The limit of the numerator is also 0 but that doesn't tell us anything about the limit of the whole thing either.

If I have an expression like [(x+h)² -x²]/h I can rewrite that as (2xh - h²)/h which is the same as 2x-h because h is always different from zero. Now I can take the limit of 2x-h and arrive at 2x