Question

\(\lim\limits_{x\rightarrow\pm\infty}\dfrac{x}{1+\left|x\right|}\)

Answer 1

TH1: \(\lim\limits_{x\rightarrow+\infty}\dfrac{x}{1+\left|x\right|}\)

\(=\lim\limits_{x\rightarrow+\infty}\dfrac{x}{1+x}\)

\(=\lim\limits_{x\rightarrow+\infty}\dfrac{1}{\dfrac{1}{x}+1}=\dfrac{1}{0+1}=1\)

TH2: \(\lim\limits_{x\rightarrow-\infty}\dfrac{x}{1+\left|x\right|}\)

\(=\lim\limits_{x\rightarrow-\infty}\dfrac{x}{1-x}\)

\(=\lim\limits_{x\rightarrow-\infty}\dfrac{1}{\dfrac{1}{x}-1}=\dfrac{1}{0-1}=-1\)