Quantify how small are the integrals $\int_0^N e^{-Nx}\binom{x}{N}dx $, as $N\to\infty$ While I was reading the Wikipedia entry for Gregory coefficients I've thought that should be very nice and fun calculate definite integrals involving binomial coefficients.
This is a simple exercise that I've thought after I did some experiments using Wolfram Alpha online calculator with codes like these:
integrate   e^(-35 x) Binomial[x,35] dx, from x=0 to x=35
integrate   e^(-200 x) Binomial[x,200] dx, from x=0 to x=200
I believe that the absolute value of integrals is small.

Question. (Being $N\geq 1$ integer) I would like to know how to quantify how small are these integrals. Does exist 
  $$\lim_{N\to\infty}\int_0^N e^{-Nx}\binom{x}{N}dx?$$ Alternatively, quantify $$ \left| \int_0^N e^{-Nx}\binom{x}{N}dx \right|$$ as $N$ tends to infinite.  Thanks in advance.


I did some calculations and it seems like your integral, call it $f(N)$, is approximately $-(-1)^N/N^3$ as $N$ gets large. You may be able to get more accurate approximation.
