Cálculo

\[\begin{array}{rcl} \lim\limits_{x \to +\infty} \sqrt{x + 1} - \sqrt{x^2 -3} & = & \lim\limits_{x \to +\infty} \frac{(\sqrt{x + 1} - \sqrt{x^2 - 3})(\sqrt{x + 1} + \sqrt{x^2 - 3})}{\sqrt{x + 1} + \sqrt{x^2 - 3}}\\ & = & \lim\limits_{x \to +\infty} \frac{x + 1 - x^2 + 3}{\sqrt{x + 1} + \sqrt{x^2 - 3}}\\ & = & \lim\limits_{x \to +\infty} \frac{x^2(\frac{1}{x} + \frac{4}{x^2} - 1)}{x(\sqrt{\frac{x + 1}{x^2}} + \sqrt{\frac{x^2 - 3}{x^2}})}\\ & = & \lim\limits_{x \to +\infty} \frac{x(\frac{1}{x} + \frac{4}{x^2} - 1)}{\sqrt{\frac{x + 1}{x^2}} + \sqrt{1 -\frac{3}{x^2}}}\\ & = & -\infty \end{array}\]