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When we have perfect elasticity, the demand curve is a horizontal line and the elasticity of demand coefficient is equal to infinity.
How do we arrive at a solution equal to infinity?
We know that
Elasticity of Demand Coefficient = Change in Quantity Demanded / Change in Price $x$
If we change price by $x$, how do we get what the change in quantity demanded is if there is no quantity demanded at the new price?
$$\epsilon_D=\frac{\% \Delta Q_D}{\% \Delta P}$$ A perfectly elastic demand curve is horizontal, meaning we quantity demanded can change by any amount without changing price (any quantity can be sold at the price corresponding the vertical intercept). Hence, for any change in quantity, $\% \Delta P=0$. While we cannot strictly divide by zero, in the limit, the ratio is infinity.
Let demand elasticity be $\varepsilon$. Then
$$\varepsilon = (\frac{p}{D(p)})\frac{\mathrm{d}D(p)}{\mathrm{d}p} \ \ \text{(1)}$$
For a perfectly elastic demand curve, $\frac{\mathrm{d}D(p)}{\mathrm{d}p} = - \infty$. Why?
$$\frac{\mathrm{d}D(p)}{\mathrm{d}p} = \lim_{\Delta p \to 0} \frac{D(p + \Delta p) - D(p)}{\Delta p}$$
Now, $D(p + \Delta p) = 0 \ \forall \ \Delta p ≥ 0$. Thus:
$$\frac{\mathrm{d}D(p)}{\mathrm{d}p} = \lim_{\Delta p \to 0} \frac{-D(p)}{\Delta p} = D(p)\lim_{\Delta p \to 0}\frac{-1}{\Delta p} \implies \frac{\mathrm{d}D(p)}{\mathrm{d}p} = -\infty $$
Hence, substituting into $\text{(1)}$, $\varepsilon = -\infty$.
