Limit and factorization

Question

I have the following very interesting homework exercise:

Let $$f(x)=\frac{x^2-4}{2-x\cdot \sqrt {x+2}+\sqrt{x+2}}$$ Find the following limit, if it exists: $$\lim_{x\to 2}f(x)$$

I understand that I need to "delete" the $x-2$ factor from both nominator and denominator and then evaluate the limit. It is obvious that $x^2-4=(x-2)\cdot (x+2)$, so the nominator is done.

So, my main problem is how to factor the denominator of $f(x)$. Any help with the factorization, without the final solution, would be appreciated. Then, after I understand how to factor it, anyone can post the full solution, even I.

Answer 1

$$\begin{align}\lim_{x\to 2}\frac{x^2-4}{2-x\sqrt{x+2}+\sqrt{x+2}}&=\lim_{x\to 2}\frac{(x-2)(x+2)}{2-(x-1)\sqrt{x+2}}\\&=\lim_{x\to 2}\frac{(x-2)(x+2)}{2-(x-1)\sqrt{x+2}}\cdot\frac{2+(x-1)\sqrt{x+2}}{2+(x-1)\sqrt{x+2}}\\&=\lim_{x\to 2}\frac{(x-2)(x+2)(2+(x-1)\sqrt{x+2})}{4-(x-1)^2(x+2)}\\&=\lim_{x\to 2}\frac{(x-2)(x+2)(2+(x-1)\sqrt{x+2})}{-(x-2)(x+1)^2}\\&=\lim_{x\to 2}\frac{(x+2)(2+(x-1)\sqrt{x+2})}{-(x+1)^2}\\&=\frac{4\cdot (2+2)}{-3^2}\end{align}$$

Answer 2

$$f(x)=\frac{x^2-4}{2-x\cdot \sqrt {x+2}+\sqrt{x+2}}\Longrightarrow$$

$$\lim_{x\to2}f(x)=\lim_{x\to2}\left(\frac{x^2-4}{2-x\cdot \sqrt {x+2}+\sqrt{x+2}}\right)=$$

$$\lim_{x\to2}\left(\frac{\frac{d}{dx}\left(x^2-4\right)}{\frac{d}{dx}\left(2-x\cdot \sqrt {x+2}+\sqrt{x+2}\right)}\right)=$$ $$\lim_{x\to2}\left(\frac{2x}{\frac{1}{2\sqrt{x+2}}-\frac{x}{2\sqrt{x+2}}-\sqrt{x+1}}\right)=$$ $$\lim_{x\to2}\left(-\frac{4x\sqrt{x+2}}{3(x+1)}\right)=$$ $$\frac{\lim_{x\to2}\left(-4x\sqrt{x+2}\right)}{\lim_{x\to2}\left(3(x+1)\right)}=$$

($3(x+1)$ is a polynomial and thus everywhere continuous):

$$\frac{\lim_{x\to2}\left(-4x\sqrt{x+2}\right)}{3(2+1)}=$$ $$\frac{\lim_{x\to2}\left(-4x\sqrt{x+2}\right)}{3(3)}=$$ $$\frac{\lim_{x\to2}\left(-4x\sqrt{x+2}\right)}{9}=$$ $$\frac{-4\lim_{x\to2}x\sqrt{\lim_{x\to2}(x+2)}}{9}=$$

($x$ is a polynomial and thus everywhere continuous):

$$\frac{-4\cdot2\sqrt{\lim_{x\to2}(x+2)}}{9}=$$

($x+2$ is a polynomial and thus everywhere continuous):

$$\frac{-4\cdot2\sqrt{(2+2)}}{9}=\frac{-4\cdot2\sqrt{4}}{9}=\frac{-4\cdot2\cdot2}{9}=\frac{-4\cdot4}{9}=\frac{-16}{9}=-\frac{16}{9}$$

So:

$$\lim_{x\to2}f(x)=\lim_{x\to2}\left(\frac{x^2-4}{2-x\cdot \sqrt {x+2}+\sqrt{x+2}}\right)=-\frac{16}{9}$$

Answer 3

Let $x+2=z^2$, then clearly $z\to 2$ as $x\to2$. After transformation$$\lim\limits_{x\to2}\frac{x^2-4}{2-x\sqrt{x+2}+\sqrt{x+2}}=$$ $$ \lim\limits_{z\to2}\frac{z^2(4-z^2)}{z^3-3z-2}=\lim\limits_{z\to2}\frac{-z^2(z+2)(z-2)}{(z-2)(1+z(z+2))}=-\frac{16}{9}.$$