Elasticity with perfectly inelastic / elastic demand When a change in price results in an infinitely large response in quantity demanded, demand is perfectly elastic. The perfectly elastic demand curve is horizontal. At price P, consumers will buy a quantity Q. If there is an increase in price, quantity demanded drops to zero due to the existence of perfect substitutes. However, when price drops, how will the PED remain infinity? Wouldn't consumers demand as much, if not more, of the product?
In the title you ask about perfectly inelastic demand; in the text it is about perfectly elastic demand. I guess you want to know about the latter. So you can skip one of the paragraphs.

Let us define PED as this absolute vale
$$\varepsilon_p = \left| \frac{d Q / Q}{dP / P} \right|,$$
(otherwise it is a non-positive value, this is just a convention).

In general, both perfectly elastic and inelastic demand are defined in the limit. As always, the math gets a little fishy when $\infty$ is involved (or if you want to divide by zero). To stay sane, it helps to think of an "infinite quanity" as it is used in economics as "very, very, very large approaching infinity" or as "as much as possible". Similarly, think of a good with an "infinite price" as a good that "nobody can buy".

With perfectly inelastic demand, price changes do not affect demanded quantity $q$. 
As an example, think of a life-saving drug. How must the price change that you demand $q'&lt;q$ instead of $q$? Well, you always demand $q$ - it doesn't happen. How can we express "it doesn't happen"? We can say that you would only consume less than $q$ if the price becomes some number that is larger than any real number. Now think about a limit approaching this benchmark case of a vertical demand curve, some sequence of very steep declining lines. Then for any $dQ/Q&lt;0$ it must be that $dP/P$ approaches infinity. Similarly, for a demanded quantity $q'&gt;q$, i.e., $dQ/Q&gt;0$, "the good has to be thrown at you". For a meaningful sequence approaching the perfectly inelastic demand, you have to ignore the restriction that prices are positive -- it must be that $dP/P \rightarrow - \infty$. Hence, $\varepsilon_p \rightarrow 0$.
Alternatively, don't think in limits, but ask the reverse question. Instead of "how must the price change that my demanded quantity changes", ask "how do I change my demanded quanity if the price changes"? The answer is, not at all: $dQ/Q =0$ for any $dP/P$, making the PED always zero. 

With perfectly elastic demand, an arbitrarily large quantity can be sold at some market price $p$. As an example, think of a \$100 bill or a (hypothetical) perfectly competitive market. "Arbitratily large" does not mean infinity, which is not a real number. 
Again ask, how does the price have to change that my demanded quantity changes? By definition, any quantity is demanded at price $P$. That is, the price does not have to change at all. Hence, $dP/P=0$ for any $dQ/Q$. Then $\varepsilon_p$ is not well-defined (you divide by zero), but you can think of a sequence of very flat declining lines that approach the horizontal demand such that $dP/P\rightarrow 0$ and $\varepsilon_p \rightarrow \infty$.
You can again also ask the reverse question: how does demanded quantity change when the price changes. For this just consider the limit again (and ignore the restricion to positive quanitites). If a \$100 bill is offered at \$99.99, demanded quantity goes to infinity. If a \$100 bill is offered at \$100.01, demanded quantity goes to minus infinity -- you would want to sell your bills. That is, $dQ/Q \rightarrow  \mbox{sign}(dP/P) (-\infty)$ and $\varepsilon_p \rightarrow \infty$.