Question

\(\lim\limits_{x\rightarrow\pm\infty}\dfrac{x+1}{\sqrt{x^2-x+1}}\)

Answer 1

TH1: \(\lim\limits_{x\rightarrow+\infty}\dfrac{x+1}{\sqrt{x^2-x+1}}\)

\(=\lim\limits_{x\rightarrow+\infty}\dfrac{1+\dfrac{1}{x}}{\sqrt{1-\dfrac{1}{x}+\dfrac{1}{x^2}}}\)

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

TH2: \(\lim\limits_{x\rightarrow-\infty}\dfrac{x+1}{\sqrt{x^2-x+1}}\)

\(=\lim\limits_{x\rightarrow-\infty}\dfrac{x+1}{-x\cdot\sqrt{1-\dfrac{1}{x}+\dfrac{1}{x^2}}}\)

\(=\lim\limits_{x\rightarrow-\infty}\dfrac{1+\dfrac{1}{x}}{-\sqrt{1-\dfrac{1}{x}+\dfrac{1}{x^2}}}=\dfrac{1+0}{-\sqrt{1-0+0}}=\dfrac{1}{-1}=-1\)