Question

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

Answer 1

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

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

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

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

\(=\dfrac{2}{\left(\sqrt{3\cdot1+1}+\sqrt{1+3}\right)\left(1^2+1+1\right)}\)

\(=\dfrac{2}{3\cdot\left(2+2\right)}=\dfrac{2}{3\cdot4}=\dfrac{2}{12}=\dfrac{1}{6}\)