I believe so. This is something I had to prove sometime back. It's a cool proof.
Let . Therefore,
Start with the binomial
Now, if we divide one of the x factors, n, in into each factor of the product , and then write
Now, let while x and lambda stay fixed.
Now, we get the famous 'e' limit:
Therefore, hence, and heretofore, we get:
and the limiting distribution becomes the Poisson distribution.