Question

lim\(\dfrac{\sqrt{x^2+4}+x-2}{x^2-x}\)(x-->0)

Answer 1

\(\lim\limits_{x\rightarrow0}\dfrac{\sqrt{x^2+4}+x-2}{x^2-x}=\lim\limits_{x\rightarrow0}\dfrac{\dfrac{\left(\sqrt{x^2+4}-2\right)\left(\sqrt{x^2+4}+2\right)}{\sqrt{x^2+4}+2}+x}{x\left(x-1\right)}\)

\(=\lim\limits_{x\rightarrow0}\dfrac{\dfrac{x^2}{\sqrt{x^2+4}+2}+x}{x\left(x-1\right)}=\lim\limits_{x\rightarrow0}\dfrac{\dfrac{x}{\sqrt{x^2+4}+2}+1}{x-1}\)

\(=\dfrac{0+1}{-1}=-1\)