Limit Rationalization

For equations like this: ${\lim }_{x\to 0}\frac{\sqrt{x+1}-1}{x}$ We cannot use Properties of limits.

Example 1

${\lim }_{x\to 0}\frac{\sqrt{x+1}-1}{x}$ Use the Conjugate to assist in rationalizing. ${\lim }_{x\to 0}\frac{\sqrt{x+1}-1}{x}\ast \frac{\sqrt{x+1}+1}{\sqrt{x+1}+1}$ ${\lim }_{x\to 0}\frac{1}{\sqrt{x+1}+1}$ $\frac{1}{\sqrt{0+1}+1}$ $\frac{1}{2}$

Example 2

Sometimes we may need to rationalize both numerator and denominator ${\lim }_{x\to 2}\frac{\sqrt{6-x}-2}{\sqrt{3-x}-1}$ ${\lim }_{x\to 2}\frac{\sqrt{6-x}-2}{\sqrt{3-x}-1}\ast \frac{\sqrt{6-x}+2}{\sqrt{6-x}+2}\ast \frac{\sqrt{3-x}+1}{\sqrt{3-x}+1}$ ${\lim }_{x\to 2}\frac{(-x+2)(\sqrt{3-x}+1)}{(-x+2)(\sqrt{6-x}+2)}$ ${\lim }_{x\to 2}\frac{(\sqrt{3-x}+1)}{(\sqrt{6-x}+2)}$ $0.5$ n r