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The commutator of position and momentum operator, $\hat{p}$ and $\hat{x}$, respectively is derived as $[\hat{x},\hat{p}]=i\hbar$. Let $\lim_{x\rightarrow x_{o}} [\hat{x},\hat{p}]=\lim_{x\rightarrow x_{o}}(\hat{x}(-i\hbar \partial/\partial x) -(-i\hbar\partial/\partial x)\hat{x})$. Then we have \begin{equation} \lim_{x\rightarrow x_{o}}((\hat{x})(-i\hbar\partial/\partial x))-\lim_{x\rightarrow x_{o}}((-i\hbar\partial/\partial x)\hat{x}) \end{equation} evaluating the limits we obtain the following:$$(x_{o}p_{o})I-(p_{o}x_{o}I)$$ since $\{x_{o},p_{o}\}\subset \mathbb{R}$ we have that the commutator vanishes for every $x_{o}\in \mathbb{R}$. Can someone explain why the limit of a commutator at any point $x_{o}\in\mathbb{R}$ vanishes but the commutator does not.

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closed as unclear what you're asking by ACuriousMind, user36790, Gert, Kyle Kanos, Norbert Schuch Dec 30 '15 at 23:27

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    $\begingroup$ $\hat{x}$ does not refer to a real number. Rather, it is a multiplication operation $\hat{x}: \psi(x) \to x\psi(x)$. Hence $\lim_{x\to x_{0}}$ doesn't make sense. $\endgroup$ – higgsss Dec 29 '15 at 6:18
  • $\begingroup$ but $\hat{x}:\mathbb{R}\times H\rightarrow H$ where $H$ is the Hilbert state space. Hence, $\hat{x}$ depends on the set of reals. $\endgroup$ – Ara Vartomian Dec 29 '15 at 6:24
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    $\begingroup$ $\hat{x}: \psi(x) \to x\psi(x)$ simultaneously acts on all $x \in \mathbb{R}$ rather than on a single value $x$. That is, it shouldn't be thought as a map of $R \times H$ onto $H$. $\endgroup$ – higgsss Dec 29 '15 at 6:36
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User 'higgsss' has already answered your question in his comments. I will just rephrase his answer here so that you can close the question in case you found the explanation satisfactory.

The simple point is that you cannot take the limit of the $\hat x$ operator, because it is an operator. That's about it.

The position operator $\hat x$ is not a map $\mathbb{R} \times \mathbb{H} \to \mathbb{H}$, but rather $\mathbb{H} \to \mathbb{H}$, just like any other hermitian observable in quantum mechanics.

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