Example 1: Square Root Limits

 
$\underset{x\to –5}{\lim }\frac{\sqrt{x+6}–1}{x+5}$ 

Step 1: Try Option 1: Plug it in

Always try plugging in the $x–\mathit{value}$ you are heading towards.

In this example we get division by zero, which means Option 1 has failed.

$\underset{x\to –5}{\lim }\frac{\sqrt{x+6}–1}{x+5}=\frac{\sqrt{(–5)+6}–1}{(–5)+5}=\frac{0}{0}$

Step 2: Rationalize the square root

Here to rationalize the square root we will need to multiply the top and bottom of the fraction by the conjugate of $\sqrt{x+6}–1$.

To multiply by the conjugate, you just switch the sign ($+/–$)of the root piece.

$\underset{x\to –5}{\lim }\frac{\sqrt{x+6}–1}{x+5}\cdot \frac{(\sqrt{x+6}+1)}{(\sqrt{x+6}+1)}$

Step 3: Do the multiplication on the piece you are trying to rationalize.

Do NOT do the multiplication between the pieces you are not trying to rationalize.

In this example we DO the multiplication on the top of the fraction using distribution (FOIL).

DO NOT dot the multiplication on the bottom of the fraction.

$\underset{x\to –5}{\lim }\frac{(\sqrt{x+6}–1)}{x+5}\cdot \frac{(\sqrt{x+6}+1)}{(\sqrt{x+6}+1)}$

$=\underset{x\to –5}{\lim }\frac{(x+6)+1\sqrt{x+6}–1\sqrt{x+6}–1}{(x+5)(\sqrt{x+6}+1)}$

$=\underset{x\to –5}{\lim }\frac{(x+6)–1}{(x+5)(\sqrt{x+6}+1)}=\underset{x\to –5}{\lim }\frac{(x+5)}{(x+5)(\sqrt{x+6}+1)}$

Step 4: Look to Factor and Cancel

You will usually find that you will get some convenient cancelation between the top and bottom of your fractions, and that that simplification is just what you needed to finish up the problem.

In this example we are able to cancel an $x+5$ on the top and bottom of the fraction.

$\underset{x\to –5}{\lim }\frac{\overline{)\left(\mathrm{(}x+5\mathrm{)}\right)}}{\overline{)\left(\mathrm{(}x+5\mathrm{)}\right)}\left(\mathrm{(}\sqrt{x+6}+1\mathrm{)}\right)}=\underset{x\to –5}{\lim }\frac{1}{\left(\mathrm{(}\sqrt{x+6}+1\mathrm{)}\right)}$

Step 5: Back to Option 1: Plug it in.

$\underset{x\to –5}{\lim }\frac{1}{\left(\mathrm{(}\sqrt{x+6}+1\mathrm{)}\right)}=\underset{x\to –5}{\lim }\frac{1}{\left(\mathrm{(}\sqrt{(–5)+6}+1\mathrm{)}\right)}=\frac{1}{\left(\mathrm{(}\sqrt{1}+1\mathrm{)}\right)}=\frac{1}{2}$

Final Result:

$\underset{x\to –5}{\lim }\frac{\sqrt{x+6}–1}{x+5}=\frac{1}{2}$

Meaning:

The overall limit as $x$  approaches $–5$ of $\frac{\sqrt{x+6}–1}{x+5}$ is $y=\frac{1}{2}$.