Discussion of Integral calculus
Moderator: msmod
Re: Fundamental paradox of Integral calculus?
There's no contradiction in saying h is not zero but lim _{h>0} (h) = 0. Recall also that in the definition of limits, the value of lim_{h>a}(f(h)) is independent of the value of f at a (f(a) need not even be defined for the limit to make sense).
Re: Fundamental paradox of Integral calculus?
a.s.h.,
As far as I know, limit_{h>0} (h) , not simply h, is used in the definition of the derivative. By the way, when I have mentioned h, I have meant h defined by limit_{h>0}.
To sum up, my opinion is that h in the definition of the derivative is not equal to zero. So terms with h are simply ignored in the final stage of the limit calculations. Therefore, limit_{h>0} (h) is defined to be equal to zero by just convention, as I suppose. Algebraically and geometrically, h cannot be zero because a derivative expresses a slope of an original function, and a slope is a difference between two yvalues divided by a difference between xvalues of an original function.
P.S. Moreover, h cannot be arbitrary because it can violate the continuity of the derivative functions, as I proved it above in this thread.
Have a nice day.
As far as I know, limit_{h>0} (h) , not simply h, is used in the definition of the derivative. By the way, when I have mentioned h, I have meant h defined by limit_{h>0}.
To sum up, my opinion is that h in the definition of the derivative is not equal to zero. So terms with h are simply ignored in the final stage of the limit calculations. Therefore, limit_{h>0} (h) is defined to be equal to zero by just convention, as I suppose. Algebraically and geometrically, h cannot be zero because a derivative expresses a slope of an original function, and a slope is a difference between two yvalues divided by a difference between xvalues of an original function.
P.S. Moreover, h cannot be arbitrary because it can violate the continuity of the derivative functions, as I proved it above in this thread.
Have a nice day.
Re: Fundamental paradox of Integral calculus?
h and lim_{h>0} (h) are not the same things. The latter is the real number 0 and the former is a dummy variable (the variable that's approaching 0 for the limit). Also, it's not a convention that lim_{h>0} (h) = 0, it's something that can be proved using the epsilondelta definition of limits.
Re: Fundamental paradox of Integral calculus?
a.s.h.,
So if F(x) = x^2 is an original function, then:
F'(x) = f(x) = limit_{h>0} ((x+h)^2  x^2)) / ((x+h)  x) = limit_{h>0} (x^2+2xh+h^2x^2) / (x+hx) =
the latter can be rewritten as:
= (x^2 + 2*x*(limit_{h>0} (h)) + (limit_{h>0} (h))^2  x^2) / (x + limit_{h>0} (h) x) =
and if limit_{h>0} (h) = 0, hence F'(x) = f(x) = (x^2 +2*x*0 + 0^2  x^2) / (x + 0  x) = (0/0)
If you tell that we cannot take the limit before the final stage, then, please explain why.
Thank you.
So if F(x) = x^2 is an original function, then:
F'(x) = f(x) = limit_{h>0} ((x+h)^2  x^2)) / ((x+h)  x) = limit_{h>0} (x^2+2xh+h^2x^2) / (x+hx) =
the latter can be rewritten as:
= (x^2 + 2*x*(limit_{h>0} (h)) + (limit_{h>0} (h))^2  x^2) / (x + limit_{h>0} (h) x) =
and if limit_{h>0} (h) = 0, hence F'(x) = f(x) = (x^2 +2*x*0 + 0^2  x^2) / (x + 0  x) = (0/0)
If you tell that we cannot take the limit before the final stage, then, please explain why.
Thank you.
Last edited by Boris Lagutin on Wed Oct 12, 2016 3:33 am, edited 2 times in total.
Re: Fundamental paradox of Integral calculus?
a.s.h.,
There is one principal point, in my opinion,  a limit when h as a length of an interval goes to zero and a limit when h as some variable (dummy as you wrote) goes to some number (zero).
I see h in the definition of the derivative as a length shrinking to zero, not as some variable going to zero. And when I treat a slope as a limit_{h>0} (the definition of the derivative), I understand that the length of the interval between two xvalues cannot be zero.
P.S. Generally speaking, all mathematics is build on convention.
Thank you.
There is one principal point, in my opinion,  a limit when h as a length of an interval goes to zero and a limit when h as some variable (dummy as you wrote) goes to some number (zero).
I see h in the definition of the derivative as a length shrinking to zero, not as some variable going to zero. And when I treat a slope as a limit_{h>0} (the definition of the derivative), I understand that the length of the interval between two xvalues cannot be zero.
P.S. Generally speaking, all mathematics is build on convention.
Thank you.
Re: Fundamental paradox of Integral calculus?
This is untrue. What we would be doing here is essentially saying that lim_{h > 0} (A(h)/B(h)) = (lim_{h>0} A(h))/(lim_{h>0} B(h)), where A(h) and B(h) are functions of h (in this case A(h) = f(x+h)  f(x) and B(h) = h, where f(t) = t^2). In other words, we are saying that the limit of a quotient is the quotient of the limits. However, this not the case if moving the limits into the numerator and denominator like this results in a 0/0. This is why we cannot do what was done above.Boris Lagutin wrote: F'(x) = f(x) = limit_{h>0} ((x+h)^2  x^2)) / ((x+h)  x) = limit_{h>0} (x^2+2xh+h^2x^2) / (x+hx) =
the latter can be rewritten as:
= (x^2 + 2*x*(limit_{h>0} (h)) + (limit_{h>0} (h))^2  x^2) / (x + limit_{h>0} (h) x)
Re: Fundamental paradox of Integral calculus?
A length shrinking to 0 or a variable going to 0 is essentially the same thing. We can think of h as the displacement from the place we are taking the derivative, and limiting this towards 0. Whether we think of it as a "length" or a "variable" doesn't affect the value of the limit.Boris Lagutin wrote: I see h in the definition of the derivative as a length shrinking to zero, not as some variable going to zero. And when I treat a slope as a limit_{h>0} (the definition of the derivative), I understand that the length of the interval between two xvalues cannot be zero.
P.S. Generally speaking, all mathematics is build on convention.
Thank you.
Regarding the comment of convention, I meant that "lim_{h> 0} (h) = 0" isn't something that's true by convention, it's something that can be proved from definitions of other concepts (limits). Things like "0! = 1" or "n choose k = 0 if k > n" are conventions.
Re: Fundamental paradox of Integral calculus?
a.s.h.,
Do you think that h is a variable or a function in the definition of the derivative (limit): F'(x) = (F(x+h)  F(x)) / h , as h goes to zero?
Thank you.
Do you think that h is a variable or a function in the definition of the derivative (limit): F'(x) = (F(x+h)  F(x)) / h , as h goes to zero?
Thank you.
Re: Fundamental paradox of Integral calculus?
It's a dummy variable for a limit.Boris Lagutin wrote:a.s.h.,
Do you think that h is a variable or a function in the definition of the derivative (limit): F'(x) = (F(x+h)  F(x)) / h , as h goes to zero?
Thank you.
Re: Fundamental paradox of Integral calculus?
a.s.h.,
Do you suppose that the limits rules for functions can be applied for h in the definition of the derivative?
Thank you.
Do you suppose that the limits rules for functions can be applied for h in the definition of the derivative?
Thank you.
Re: Fundamental paradox of Integral calculus?
Which rules are you referring to? Remember, we can't move the limit into the numerator and denominator, because this results in 0/0, and so the limit quotient rule doesn't apply.Boris Lagutin wrote:a.s.h.,
Do you suppose that the limits rules for functions can be applied for h in the definition of the derivative?
Thank you.
Re: Fundamental paradox of Integral calculus?
a.s.h.,
I mean the laws of limits like the product law, sum etc. By the way, h can be only the most minimal value, in other words, h should be constant, in according to my proof of the paradox. If h is not the most minimal value, then the continuity of the derivative functions may be violated, in according to my proof of the paradox.
I must say that this discussion is great and exciting for me. I even cannot normally concentrate on my studies. I need to complete some courses yet. So I need to take "studies break"
Once again, thank you and Joe so much for such a good discussion!
Have a nice day.
I mean the laws of limits like the product law, sum etc. By the way, h can be only the most minimal value, in other words, h should be constant, in according to my proof of the paradox. If h is not the most minimal value, then the continuity of the derivative functions may be violated, in according to my proof of the paradox.
I must say that this discussion is great and exciting for me. I even cannot normally concentrate on my studies. I need to complete some courses yet. So I need to take "studies break"
Once again, thank you and Joe so much for such a good discussion!
Have a nice day.
Last edited by Boris Lagutin on Wed Oct 12, 2016 6:59 am, edited 2 times in total.
Re: Fundamental paradox of Integral calculus?
We cannot use the limit quotient rule here, since it results in a zero denominator. The quotient rule (i.e. the rule lim_{x>a} (A(x)/B(x)) = (lim_{x>a} (A(x)))/(lim_{x>a} (B(x)))) can only be used for limits where the denominator does not have 0 as limit, i.e. where lim_{x>a} (B(x)) ≠ 0.
Re: Fundamental paradox of Integral calculus?
Dear a.s.h.,
Let me ask you one question. Assume F(x)=x^2 and F'(x)=f(x) (please, see the attachment). May one get a slope of the original function F(x) at some value of x if a difference between the original function two values of y is equal to zero and a difference between the original function two values of x is equal to zero?
P.S. My understanding of h in the definition of the derivative is that h goes to zero but not zero. In this way, the limit as h goes to zero is NOT undetermined on this stage: lim_{h>0} ((x^2+2xh+h^2)x^2) / ((x+h)x). Therefore, we can continue to calculate this limit and simply ignore terms with h at the final stage because such terms are very very small. However, terms with h cannot be ignored in infinite series (see my comments above), in my opinion. It's interesting what Isaac Newton would say about it.
Have a nice day.
Let me ask you one question. Assume F(x)=x^2 and F'(x)=f(x) (please, see the attachment). May one get a slope of the original function F(x) at some value of x if a difference between the original function two values of y is equal to zero and a difference between the original function two values of x is equal to zero?
P.S. My understanding of h in the definition of the derivative is that h goes to zero but not zero. In this way, the limit as h goes to zero is NOT undetermined on this stage: lim_{h>0} ((x^2+2xh+h^2)x^2) / ((x+h)x). Therefore, we can continue to calculate this limit and simply ignore terms with h at the final stage because such terms are very very small. However, terms with h cannot be ignored in infinite series (see my comments above), in my opinion. It's interesting what Isaac Newton would say about it.
Have a nice day.
 Attachments

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Re: Fundamental paradox of Integral calculus?
If the difference between the xvalues was 0, we'd get 0/0 for the difference quotient, which is indeterminate. So we instead look at a limit as the difference between the xvalues (h) tends to 0.