# Find the following limits or state that they do not exist. Assume a, b, c, and k

Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers.
$\underset{x\to 9}{lim}\frac{\sqrt{x}-3}{x-9}$
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cyhuddwyr9
To find $\underset{x\to 9}{lim}\frac{\sqrt{x}-3}{x-9}$
We rationalize the numerator as
$\underset{x\to 9}{lim}\frac{\sqrt{x}-3}{x-9}×\frac{\sqrt{x}+3}{\sqrt{x}+3}$
$=\underset{x\to 9}{lim}\frac{x+3\sqrt{x}-3\sqrt{x}-9}{\left(x-9\right)\left(\sqrt{x}+3\right)}$
$=\underset{x\to 9}{lim}\frac{x-9}{\left(x-9\right)\left(\sqrt{x}+3\right)}$
$=\underset{x\to 9}{lim}\frac{1}{\sqrt{x}+3}$
$=\frac{1}{\sqrt{9}+3}$
$=\frac{1}{6}$
Therefore, $\underset{x\to 9}{lim}\frac{\sqrt{x}-3}{x-9}=\frac{1}{6}$
Jeffrey Jordon