How Does the Limit Process Simplify the Expression x^2-36?

[fermio]

Homework Statement

$$\lim_{x\to 6}\frac{\sqrt{x+3}-3}{x-6}$$

Homework Equations

Answer is 1/6.

The Attempt at a Solution

$$\lim_{x\to 6}\frac{\sqrt{x+3}-3}{x-6}=\lim_{x\to 6}\frac{(x+3-9)(x+6)}{(x^2-36)(\sqrt{x+3}+3)}$$

 

[Mathdope]

Homework Statement

$$\lim_{x\to 6}\frac{\sqrt{x+3}-3}{x-6}$$

Homework Equations

Answer is 1/6.

The Attempt at a Solution

$$\lim_{x\to 6}\frac{\sqrt{x+3}-3}{x-6}=\lim_{x\to 6}\frac{(x+3-9)(x+6)}{(x^2-36)(\sqrt{x+3}+3)}$$

Look more carefully at your attempt at a solution. You actually have it.

 

[arildno]

Do not bother with expanding with (x+6) as well:
$$\frac{\sqrt{x+3}-3}{(x-6)}=\frac{x+3-9}{(x-6)(\sqrt{x+3}+3)}=\frac{(x-6)}{(x-6)(\sqrt{x+3}+3)}$$

 

[fermio]

"Look more carefully at your attempt at a solution. You actually have it."
I don't see.
$$\lim_{x\to 6}\frac{\sqrt{x+3}-3}{x-6}=\lim_{x\to 6}\frac{(x+3-9)(x+6)}{(x^2-36)(\sqrt{x+3}+3)}=\lim_{x\to 6}\frac{x^2+6x-6x-36}{x^2\sqrt{x+3}+3x^2-36\sqrt{x+3}-108}$$

I get it.
$$\lim_{x\to 6}\frac{\sqrt{x+3}-3}{x-6}=\lim_{x\to 6}\frac{x+3-9}{(x-6)(\sqrt{x+3}+3)}=\lim_{x\to 6}\frac{1}{\sqrt{x+3}+3}=\frac{1}{6}$$

 

[Rainbow Child]

How can you write $x^2-36$?