Solve H = sum _ k = 1 ^ M limit (as p _ { k } approaches 0) of P _ { k}log_{2}(1/p_{k})

$$H = \sum _ { k = 1 } ^ { M } \lim _ { p _ { k } \rightarrow 0 } P _ { k } \log _ { 2 } ( \frac { 1 } { p _ { k } } )$$

Solve for H

$$H=M\left(\lim_{p_{k}\to 0}P_{k}\log_{2}\left(\frac{1}{p_{k}}\right)\right)$$

Solve for M

$$\left\{\begin{matrix}M=\frac{H}{\lim_{p_{k}\to 0}P_{k}\log_{2}\left(\frac{1}{p_{k}}\right)}\text{, }&p_{k}=0\text{ or }\left(p_{k}\neq 1\text{ and }P_{k}\neq 0\right)\\M\in \mathrm{C}\text{, }&\left(P_{k}=0\text{ or }p_{k}=1\right)\text{ and }H=0\text{ and }p_{k}\neq 0\end{matrix}\right.$$