Question

For an infinitely differentiable function \(f : \mathbb R \rightarrow \mathbb R\) and arbitrary choice of \(a \in \mathbb R\), is the following statement true? \[\frac{\partial}{\partial x} \left( \lim_{x \rightarrow a} f(x) \right) = \lim_{x \rightarrow a} \frac{\partial f}{\partial x}(x)\]

Answer 1

\[\lim_{x\rightarrow a} f(x)\] is \(x\)-free.

\[\frac{\partial}{\partial x}\left(\lim_{x\rightarrow a} f(x)\right)\] is always equal to ?

Answer 2

Ah, I see what you are pointing at... What I'm trying to express I guess is not exactly what I wrote.

Answer 3

But you are definitely right haha.

Answer 4

What made me think to this (which i now realize is false), is I was looking for a simple way to show \[\lim_{x \rightarrow \infty} f(x) = 0 \Rightarrow \lim_{x\rightarrow \infty} \frac{\partial f}{\partial x}(x) =0.\]