Question

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

Answer 1

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

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

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

\(=-\infty\) vì \(\left\{{}\begin{matrix}\lim\limits_{x\rightarrow-\infty}-\sqrt{1+\dfrac{1}{x^2}}=-1< 0\\\lim\limits_{x\rightarrow-\infty}1+\dfrac{1}{x}-\sqrt{1+\dfrac{1}{x^2}}=1+0-\sqrt{1+0}=0\end{matrix}\right.\)