Question

lim^{\(\dfrac{\sqrt{4x^2+5}-3}{2x-2}\)(x-->1)}

Answer 1

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

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

\(=\lim\limits_{x\rightarrow1}\dfrac{2\left(x+1\right)}{\sqrt{4x^2+5}+3}=\dfrac{4}{6}=\dfrac{2}{3}\)