In Modern Quantum Mechanics by Sakurai, at page 46 while deriving commutator of translator operator with position operator, he uses $\left| x+dx\right\rangle \simeq \left| x \right\rangle$. But for every $\epsilon > 0$ $\langle x+ \epsilon \left| x \right\rangle  = 0$. Therefore this limiting process $\lim_{\epsilon \rightarrow 0} \left| x+ \epsilon \right\rangle = \left| x \right\rangle$ does not make sense for me. I couldn't derive commutator relation without using these. Thanks for any help.