Swapping partial derivatives and limits

Asked Jun 11 '22 at 00:36

Active Jun 11 '22 at 00:36

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What conditions does a function have to meet for the following to be true?

$$\displaystyle\lim_{x\to 0}\frac{\partial}{\partial x}f(x,y)=\frac{\partial}{\partial x}\lim_{x\to 0}f(x,y)$$

I've looked both on MSE and tried googling this question as well but I can't find a good answer. Any explanation or resources you could point me to would be greatly appreciated!

asked Jun 11 '22 at 00:36

Does this give you what you need? – Stephen Donovan Jun 11 '22 at 00:40
@StephenDonovan yes, I believe it does thank you! – Jun 11 '22 at 01:07
4

There’s something seriously wrong. The expression on the right is always $0$, since the limit is a function of $y$ alone. – Ted Shifrin Jun 11 '22 at 04:15

Swapping partial derivatives and limits

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