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**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 $\displaystyle lim_{t->0}(\frac{d}{dx}t^{-1}*sin(t*x))=lim_{t->0}(cos(t*x))=1$