Poisson becomes normal as \(\mu \rightarrow \infty\text{.}\) Consider \(\mu = 20\text{.}\) Then, \(\sigma^2 = \mu = 20\text{.}\)
Using the Poisson formulas, for example,
\begin{equation*}
P( X = 19 ) = \frac{20^{19} e^{-20}}{19!} \approx 0.08883
\end{equation*}
Using the normal distribution,
\begin{align*}
P( X = 19 ) & = P( 18.5 \lt X \lt 19.5) \\
& \approx normalcdf(18.5,19.5,20,sqrt(20)) \\
& = 0.08683
\end{align*}
Again, these are very close.