Do limit and differential operators commute?

[2sin54]

In general I'm wondering if

$$\lim_{x\to0} \left[\frac{d}{dy} \frac{d}{dx} f(x,y)\right] = \frac{d}{dy} \left[\lim_{x\to0} \frac{d}{dx} f(x,y)\right]$$

holds true for all f(x,y). Thanks.

 

[jedishrfu]

I think this is true provided the mixed partial derivatives of function f(x,y) are continuous at x=0 and whatever y=value.

http://www.math.ucsd.edu/~mradclif/teaching/Math10C/LectureNotes/second_order_partial_derivatives.pdf

Are you trying to prove this?

 

[2sin54]

I think this is true provided the mixed partial derivatives of function f(x,y) are continuous at x=0 and whatever y=value.

http://www.math.ucsd.edu/~mradclif/teaching/Math10C/LectureNotes/second_order_partial_derivatives.pdf

Are you trying to prove this?

No, not exactly. My f(x,y) is a huge compound function and it's a hassle to differentiate with respect to both variables and then take the limit, as opposed to differentiating by x, setting the limit and then differentiating (the now simpler function) by y.