$\lim\limits_{x\to 1} \dfrac{\sqrt{x+24}-5}{x-1}$
$=\lim\limits_{x\to 1} \dfrac{\big(\sqrt{x+24}-5\big)\big(\sqrt{x+24}+5\big)}{(x-1)\big(\sqrt{x+24}+5\big)}$
$=\lim\limits_{x\to 1} \dfrac{x-1}{(x-1)\big(\sqrt{x+24}+5\big)}$
$=\lim\limits_{x\to 1} \dfrac1{\sqrt{x+24}+5}$
$=\dfrac1{\sqrt{1+24}+5}=\dfrac1{10}$