Chọn phương án B.

\(\begin{aligned}
\lim\limits_{x\to6}\dfrac{\sqrt{x+3}-3}{x-6}&=\lim\limits_{x\to6}\dfrac{\left(\sqrt{x+3}-3\right)\left(\sqrt{x+3}+3\right)}{(x-6)\left(\sqrt{x+3}+3\right)}\\
&=\lim\limits_{x\to 6}\dfrac{(x+3)-9}{(x-6)\left(\sqrt{x+3}+3\right)}\\
&=\lim\limits_{x\to 6}\dfrac{1}{\sqrt{x+3}+3}\\
&=\dfrac{1}{\sqrt{6+3}+3}=\dfrac{1}{6}.
\end{aligned}\)