求 $\lim\limits_{x\rightarrow+\infty}x(\sqrt{x^2+1}-x)$.
求 $\lim_{x\rightarrow+\infty}x(\sqrt{x^2+1}-x)$.
Answer
问题及解答
1
\[
\begin{split}
&\lim_{x\rightarrow+\infty}x(\sqrt{x^2+1}-x)\\
=&\lim_{x\rightarrow+\infty}\frac{x(\sqrt{x^2+1}-x)(\sqrt{x^2+1}+x)}{\sqrt{x^2+1}+x}\\
=&\lim_{x\rightarrow+\infty}\frac{x}{\sqrt{x^2+1}+x}\\
=&\lim_{x\rightarrow+\infty}\frac{1}{\sqrt{1+\frac{1}{x^2}}+1}\\
=&\frac{1}{2}.
\end{split}
\]