$\lim\limits_{x\to 1}\dfrac{ \sqrt{2x+7}-3+\sqrt{8x+17}-5}{1-x}$
$=\lim\limits_{x\to 1}\Big( \dfrac{ 2x+7-9}{-(x-1)(\sqrt{2x+7}+3)} +\dfrac{8x+17-25}{-(x-1)(\sqrt{8x+17}+5}\Big)$
$=\lim\limits_{x\to 1}\Big( \dfrac{-2}{\sqrt{2x+7}+3}-\dfrac{8}{\sqrt{8x+17}+5}\Big)$
$=\dfrac{-17}{15}$