[tex]\lim\limits_{x\to 1}\dfrac{\sqrt{x+3}+\sqrt{7x+9}-6}{\sqrt{x}+\sqrt{2x-1}-2}\overset{\frac{0}{0}\,(L'H)}{=}\lim\limits_{x\to 1}\dfrac{\frac{1}{2\sqrt{x+3}}+\frac{7}{2\sqrt{7x+9}}-0}{\frac{1}{\sqrt{2x}}+\frac{2}{2\sqrt{2x-1}}-0}=\\ \\ = \dfrac{\frac{1}{2\sqrt{1+3}}+\frac{7}{2\sqrt{7+9}}}{\frac{1}{2}+\frac{2}{2\sqrt{2-1}}} = \dfrac{\frac{1}{2\cdot 2}+\frac{7}{2\cdot 4}}{\frac{1}{2}+7} = \dfrac{2+7}{4+8}=\dfrac{9}{12}=\mathbf{\dfrac{3}{4}}[/tex]
