$\lim\limits_{x\to 2}\dfrac{ \sqrt{2x+5}-3}{x^2-4}$
$=\lim\limits_{x\to 2}\dfrac{2x+5-9}{(x^2-4)(\sqrt{2x+5}+3)}$
$=\lim\limits_{x\to 2}\dfrac{2(x-2)}{(x-2)(x+2)(\sqrt{2x+5}+3)}$
$=\lim\limits_{x\to 2}\dfrac{2}{(x+2)(\sqrt{2x+5}+3)}$
$=\dfrac{2}{(2+2)(3+3)}$
$=\dfrac{1}{12}$