<s>[INST] Simplify a radical expression for a limit I'm supposed to find the slope of the tangent of the function $$D(p)=\frac{20}{\sqrt{p-1}}$$ at $p=5$.
Using the definition of slope of a tangent at a point, I get
$$\lim_{h \to 0}\frac{\frac{20}{\sqrt{(5+h)-1}}-10}{h}$$
Which, using "regular" (high school) algebra I simplified to $$\frac{10(2-\sqrt{h+4})}{h\sqrt{h+4}}.$$
Rationalizing the numerator or denominator doesn't help in terms of removing $h$ as a factor in the denominator of the slope. How do you find the limit algebraically in this case (or can you)?
 [/INST] Multiplying by the conjugate, we obtain:
$$ \begin{align*}
\dfrac{10(2-\sqrt{h+4})}{h\sqrt{h+4}}
&=\dfrac{10(2-\sqrt{h+4})}{h\sqrt{h+4}} \cdot \dfrac{2+\sqrt{h+4}}{2+\sqrt{h+4}}\\
&=\dfrac{10(4-(h+4))}{h\sqrt{h+4}(2+\sqrt{h+4})}\\
&=\dfrac{-10h}{h\sqrt{h+4}(2+\sqrt{h+4})}\\
&=\dfrac{-10}{\sqrt{h+4}(2+\sqrt{h+4})}\\
\end{align*} $$
So taking the limit as $h\to0$ yields:
$$
\dfrac{-10}{\sqrt{0+4}(2+\sqrt{0+4})}=\dfrac{-10}{2(2+2)}=\dfrac{-5}{4}
$$
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