differentiating limit statements

**JeffN12345** · January 9th 2010, 08:40 AM

if f(x,t) is differentiable on a subset S of the real plane, under what conditions does
$\frac{d}{dx}(lim_{t->a}f(x,t))=lim_{t->a}(\frac{d}{dx}f(x,t))$

e.g: i) $\frac{d}{dx}(lim_{t->a}((x-t)^2)=\frac{d}{dx}(x-a)^2=2(x-a)$
and $lim_{t->a}(\frac{d}{dx}(x-t)^2)=lim_{t->a}2(x-t)=2(x-a)$
ii) $\frac{d}{dx}(lim_{t->0}(t^{-1}*sin(t*x)))=\frac{d}{dx}x=1$
and $lim_{t->0}(\frac{d}{dx}t^{-1}*sin(t*x))=lim_{t->0}(cos(t*x))=1$

**Drexel28** · January 9th 2010, 02:28 PM

Originally Posted by JeffN12345

if f(x,t) is differentiable on a subset S of the real plane, under what conditions does
$\frac{d}{dx}(lim_{t->a}f(x,t))=lim_{t->a}(\frac{d}{dx}f(x,t))$

e.g: i) $\frac{d}{dx}(lim_{t->a}((x-t)^2)=\frac{d}{dx}(x-a)^2=2(x-a)$
and $lim_{t->a}(\frac{d}{dx}(x-t)^2)=lim_{t->a}2(x-t)=2(x-a)$
ii) $\frac{d}{dx}(lim_{t->0}(t^{-1}*sin(t*x)))=\frac{d}{dx}x=1$
and $lim_{t->0}(\frac{d}{dx}t^{-1}*sin(t*x))=lim_{t->0}(cos(t*x))=1$

Let us restate this, since it is the same thing as "When is $\frac{d}{dx}\lim_{n\to\infty}f_n(x)=\lim_{n\to\inf ty}\frac{d}{dx}f_n(x)$ ?". The answer is quite strict. We need that $f_n(x)$ is uniformly convergent, $f_n(x)$ is differentiable for all $n$ , and that $f_n'(x)$ is uniformly convergent.

**JeffN12345** · January 9th 2010, 02:38 PM

considering this, under which circumstances is $f(x)=\lim_{n->\infty}f_n(x)$ where $f_n(x)=\Sigma_{k=1}^n(\frac{1}{n}(f(\frac{kx-(n-k)a}{n})+\frac{k(x-a)}{n}f'(\frac{kx+(n-k)a}{n})))$
(I would like if f(x) and f'(x) were uniformly continuous over some subest S of R this would be enough. Is this the case?)

**Drexel28** · January 9th 2010, 02:39 PM

Originally Posted by JeffN12345

considering this, under which circumstances is $f(x)=\lim_{n->\infty}\Sigma_{k=1}^n(\frac{1}{n}(f(\frac{kx-(n-k)a}{n})+\frac{k(x-a)}{n}f'(\frac{kx+(n-k)a}{n})))$

I remember your last thread Jeff. This gets even more complicated. Write exactly you want to do.

**JeffN12345** · January 9th 2010, 02:52 PM

im considering the x-derivative of the left-riemann-sum of f(t) over the n-regular partition of the the interval [a,x] and how it should be converge to f(x)
i would like this to work for any a and x for which the function is riemann-integrable on [a,x]

**Drexel28** · January 9th 2010, 03:00 PM

Originally Posted by JeffN12345

im considering the x-derivative of the left-riemann-sum of f(t) over the n-regular partition of the the interval [a,x] and how it should be converge to f(x)
i would like this to work for any a and x for which the function is riemann-integrable on [a,x]

What is your background? What math do you know? I mean, do you know the concept of uniform convergence? I don't ask this to be condescending but because it will dictate how I answer your question.

**JeffN12345** · January 9th 2010, 03:09 PM

im in first year, second term at waterloo and im taking calculus 2 now. We just finished defining an integral and the conditions for riemann-integrability. I think we will learn uniform convergence later this term, but i just read what it is in my textbook after you mentioned it.

**Drexel28** · January 9th 2010, 03:13 PM

Originally Posted by JeffN12345

im in first year, second term at waterloo and im taking calculus 2 now. We just finished defining an integral and the conditions for riemann-integrability. I think we will learn uniform convergence later this term.

That amazes me. You are in a course learning about Riemann integrability (I assume that means Riemann-Stieltjes integrability for me), but you haven't learned about uniform convergence?

Also, don't most Calc II courses do stuff along these lines $\int x^n\text{ }dx=\frac{x^{n+1}}{n+1}+C\quad n\in\mathbb{N}$

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