Limits involving Rationalization

When dealing with sums or differences of square roots, we sometimes need to rationalize the expression. The key fact is that $$(\sqrt{A} + \sqrt{B})(\sqrt{A} - \sqrt{B}) = A-B,$$ so $$\sqrt{A} - \sqrt{B} = \frac{A-B}{\sqrt{A}+\sqrt{B}},$$ no matter what $A$ and $B$ are (as long as they are non-negative). For example, instead of working with $\sqrt{x+1}-\sqrt{x}$, it is sometimes easier to work with $$\left(\sqrt{x+1}-\sqrt{x}\right)\cdot\frac{\sqrt{x+1}+\sqrt{x}}{\sqrt{x+1}+\sqrt{x}}=\frac{x+1-x}{\sqrt{x+1}+\sqrt{x}} = \frac{1}{\sqrt{x+1}+\sqrt{x}}.$$ This trick works both with square roots in the numerator and with square roots in the denominator: $$\frac{1}{\sqrt{A} - \sqrt{B}} = \frac{\sqrt{A}+\sqrt{B}}{A-B};$$ and also in cases like $$ A\pm \sqrt{B}, \text{ or } \frac{1}{A \pm \sqrt{B}}. $$ The following video shows how to use this trick to get limits.