$\displaystyle\lim_{x \to 1} \dfrac{f(x)-16}{x-1}=24\\ \Rightarrow f(x)-16=(x-1)g(x)\\ \Rightarrow \displaystyle\lim_{x \to 1} g(x)=g(1)=24\\ f(x)-16=(x-1)g(x); x=1\Rightarrow f(1)-16=0 \Leftrightarrow f(1)=16\\ I=\displaystyle\lim_{x \to 1} \dfrac{f(x)-16}{(x-1)\sqrt{2f(x)+4}+6}\\ =\displaystyle\lim_{x \to 1} \dfrac{g(x)}{\sqrt{2f(x)+4}+6}\\ =\displaystyle\lim_{x \to 1} \dfrac{g(1)}{\sqrt{2f(1)+4}+6}\\ =\dfrac{g(1)}{\sqrt{2f(1)+4}+6}\\ =\dfrac{24}{\sqrt{2.16+4}+6}\\ =2$
