How do I rationalize this? Calc1

Question

Could someone walk me through this?

I need to Rationalize the numerator of the formula $$ \frac{\sqrt{\dfrac{16-(2+h)^2}{16}} - \dfrac{\sqrt3}{2}}{(2+h)-2} $$ to rewrite the expression so that it looks like $f(h)/g(h)$, subject to these two conditions:

1. the numerator $f(h)$ defines a line of slope $-1$;
2. the function $f(h)/g(h)$ is defined for $h=0$.

When you do this, what are $f(h)$ and $g(h)$?

Answer 1

Hint

$$\left(\sqrt{\dfrac{16-(2+h)^2}{16}}-\frac{\sqrt{3}}{2}\right)\left(\sqrt{\dfrac{16-(2+h)^2}{16}}+\frac{\sqrt{3}}{2}\right)=\frac{16-(2+h)^2}{16}-\frac{12}{16}=\\ =\frac{4-(2+h)^2}{16}=-\frac{h(h+2)}{16}$$

so,

$$\frac{\sqrt{\dfrac{16-(2+h)^2}{16}} - \dfrac{\sqrt3}{2}}{(2+h)-2}\cdot \frac{\sqrt{\dfrac{16-(2+h)^2}{16}}+\frac{\sqrt{3}}{2}}{\sqrt{\dfrac{16-(2+h)^2}{16}}+\frac{\sqrt{3}}{2}}$$

Can you finish?