Question

\(\lim\limits_{x\rightarrow1^+}\dfrac{\sqrt{x+3}-\sqrt{3x+1}}{\sqrt{x-1}}\)

Answer 1

\(\lim\limits_{x\rightarrow1^+}\dfrac{\sqrt{x+3}-\sqrt{3x+1}}{\sqrt{x-1}}\)

\(=\lim\limits_{x\rightarrow1^+}\left(\dfrac{x+3-3x-1}{\sqrt{x+3}+\sqrt{3x+1}}\cdot\dfrac{1}{\sqrt{x-1}}\right)\)

\(=\lim\limits_{x\rightarrow1^+}\left(\dfrac{-2x+2}{\left(\sqrt{x-1}\right)\left(\sqrt{x+3}+\sqrt{3x+1}\right)}\right)\)

\(=\lim\limits_{x\rightarrow1^+}\left(\dfrac{-2\cdot\sqrt{x-1}}{\sqrt{x+3}+\sqrt{3x+1}}\right)\)

\(=\dfrac{-2\cdot\sqrt{1-1}}{\sqrt{1+3}+\sqrt{3\cdot1+1}}=0\)