Answer
$$0$$
Work Step by Step
We find the limit as follows:
\begin{align*}
\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+1}-x \right)&=\lim _{x \rightarrow \infty} \frac{1}{\sqrt{x^{2}+1}+x}\\
&=\lim _{x \rightarrow \infty} \frac{1/x^2}{\sqrt{x^{2}/x^2+1/x^2}+x/x^2}\\
&= \lim _{x \rightarrow \infty} \frac{1/x^2}{\sqrt{1+1/x^2}+1/x}\\
&=\frac{0}{1}=0
\end{align*}
