一道有理化的题

求 $\lim\limits_{x \to \infty}x-\sqrt{x^2-1}$

$$
\begin{aligned}
&\lim_{x \to \infty} x-\sqrt{x^2-1} \\
=&\lim_{x \to \infty} \frac{\left(x-\sqrt{x^2-1}\right)\left(x+\sqrt{x^2-1}\right)}{x+\sqrt{x^2-1}} \\
=&\lim_{x \to \infty} \frac{x^2-x^2+1}{x+\sqrt{x^2-1}} \\
=&\lim_{x \to \infty} \frac{1}{x+\sqrt{x^2-1}} \\
=&0
\end{aligned}
$$