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I was reading Introduction to Probability Models 11th Edition and saw this proof of why Poisson Distribution is the approximation of Binomial Distribution when n is large and p is small:

I can understand most part of the proof except for this equation:

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I really don’t remember where it comes from, could anybody explain this to me? Thanks!.

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Well, this is a basic fact of the exponential function $e^x$.

One definition of $e$ is the limit $\lim_{n\to\infty}(1+\frac1n)^n$. By a monotonicity argument one can prove $\lim_{x\to\infty}(1+\frac1x)^x=e$ where $x$ now ranges the real numbers.

Also note that $1-\frac1x=\frac{x-1}x=1/\frac x{x-1}=1/(1+\frac1y)$ where $y=x-1$.

So, $\lim_{x\to\infty}(1-\frac1x)^x=\lim_{y\to\infty}(1+\frac1y)^{-(y+1)}=e^{-1}$ (using $(1+\frac1y)\overset{y\to\infty}\to 1$ for the extra $+1$ in the exponent).

From here, assuming $\lambda>0$,

$$\begin{aligned}e^{-\lambda}=(e^{-1})^y &= \lim_{x\to\infty}(1-\frac1x)^{\lambda x} =&\to_{\ z:=\lambda x}

\\

&= \lim_{z\to\infty}(1-\frac\lambda z)^z\,.

\end{aligned} $$

In consequence, we have this limit for every sequence $z_n\to\infty$ written in place of $z$ and limiting on the natural $n\to\infty$. In particular, this also holds for $z_n=n$.

Note that we had to take the turnaround for arbitrary real numbers instead of integers only because of the exponent $\lambda$.

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