Question

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

Answer 1

\(\lim\limits_{x\rightarrow1}\dfrac{\sqrt{3x^2+2x-1}-2}{x^2-1}=\lim\limits_{x\rightarrow1}\dfrac{3x^2+2x-5}{\left(x-1\right)\left(x+1\right)\left(\sqrt{3x^2+2x-1}+2\right)}\)

\(=\lim\limits_{x\rightarrow1}\dfrac{\left(x-1\right)\left(3x+5\right)}{\left(x-1\right)\left(x+1\right)\left(\sqrt{3x^2+2x-1}+2\right)}\)

\(=\lim\limits_{x\rightarrow1}\dfrac{3x+5}{\left(x+1\right)\left(\sqrt{3x^2+2x-1}+2\right)}=\dfrac{8}{2\left(\sqrt{4}+2\right)}=1\)