1. ## differentiating limit statements

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

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

2. Originally Posted by JeffN12345
if f(x,t) is differentiable on a subset S of the real plane, under what conditions does
$\displaystyle \frac{d}{dx}(lim_{t->a}f(x,t))=lim_{t->a}(\frac{d}{dx}f(x,t))$

e.g: i) $\displaystyle \frac{d}{dx}(lim_{t->a}((x-t)^2)=\frac{d}{dx}(x-a)^2=2(x-a)$
and $\displaystyle lim_{t->a}(\frac{d}{dx}(x-t)^2)=lim_{t->a}2(x-t)=2(x-a)$
ii) $\displaystyle \frac{d}{dx}(lim_{t->0}(t^{-1}*sin(t*x)))=\frac{d}{dx}x=1$
and $\displaystyle lim_{t->0}(\frac{d}{dx}t^{-1}*sin(t*x))=lim_{t->0}(cos(t*x))=1$
Let us restate this, since it is the same thing as "When is $\displaystyle \frac{d}{dx}\lim_{n\to\infty}f_n(x)=\lim_{n\to\inf ty}\frac{d}{dx}f_n(x)$?". The answer is quite strict. We need that $\displaystyle f_n(x)$ is uniformly convergent, $\displaystyle f_n(x)$ is differentiable for all $\displaystyle n$, and that $\displaystyle f_n'(x)$ is uniformly convergent.

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

4. Originally Posted by JeffN12345
considering this, under which circumstances is $\displaystyle f(x)=\lim_{n->\infty}\Sigma_{k=1}^n(\frac{1}{n}(f(\frac{kx-(n-k)a}{n})+\frac{k(x-a)}{n}f'(\frac{kx+(n-k)a}{n})))$
I remember your last thread Jeff. This gets even more complicated. Write exactly you want to do.

5. 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]

6. 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.

7. 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.

8. 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 $\displaystyle \int x^n\text{ }dx=\frac{x^{n+1}}{n+1}+C\quad n\in\mathbb{N}$