Let us look at the behaviour of the frequency plots of random variables $X_k$ that are distributed according to the Negative Binomial distribution $NB(k;p)$ for a fixed $p$ and various $k$. Recall that:

$$P(X_k=r)=\left(\genfrac{}{}{0ex}{}{k+r-1}{r}\right)p^k(1-p{)}^r$$

sage: def nbinom(k,p,r):
...       return N(binomial(k+r-1,r)*p^k*(1-p)^r)

In order to compare them we create a sequence with $p=1/2$ and varying $k$.

sage: p=1/2
sage: nbinoms=[bar_chart(map(lambda r: nbinom(k,p,r), range(20)),color='gray', width=0.2) for k in range(1,20)]

We then animate this sequence.

sage: nbanim=animate(nbinoms,xmin=0,xmax=20,ymin=0,ymax=0.5,figsize=[6,4])

There is no discernable limiting behavior except that the mode moves to the right (keeps increasing) as does the mean.

sage: nbanim.show(delay=70)

Negative Binomial spreads and flattens as k increases

Therefore, let us fix $k$. Let us introduce a new variable $n$ and "increase the frequency of checks" as follows. We replace $r$ by $nt$ and $p$ by $c/n$ as we did for the Binomial distribution.

In other words, we are considering a sequence of random variables $Y_n$ so that $n(Y_n)$ is distributed according to $NB(k;c/n)$.

We then get the formula $P(Y_n=t)=\left(\genfrac{}{}{0ex}{}{k+tn-1}{tn}\right)(c/n{)}^k(1-c/n{)}^{tn}$.

In this case, we expect the limit to be a density, so we should consider the frequencies to be divided by the interval size ( $1/n$) and we must also use $1/n$ as the step size.

The limiting density is the Poisson density $\frac{c^k{t}^{k-1}}{(k-1)!}{e}^{-ct}$.

sage: c=0.9;k=2;var('t')
sage: poden=plot(c^k*t^(k-1)*exp(-c*t)/factorial(k-1),(t,0,10),color='orange')
sage: nseq=[poden+point(map(lambda t: (t,n*nbinom(k,c/n,t*n)), srange(0,10,1/n)),color='gray') for n in range(1,30)]

sage: nanim=animate(nseq,xmin=0,xmax=10,ymin=0,ymax=c/2,figsize=[6,4])

sage: nanim.show(delay=70)

Negative Binomial to Poisson Density