Hi. I need to find the limit as m goes to infinity of this function:

$\displaystyle [\frac{m!m!}{(m+k)!(m-k)!}]^m$. It might be sort of hard to read but it's that whole fraction raised to the m power.

My teacher first suggested trying to compute the limit as m goes to infinity of $\displaystyle [\frac{m}{m+k}]^m$. I still wasn't exactly sure even how to do that, but tried using L'Hospital and changed it to the limit as m goes to infinity of $\displaystyle \frac{(ln(m) + 1)m^m}{(ln(m+k)+\frac{m}{m+k})(m+k)^m}$, which honestly made me even more confused.

THEN, she said to use the fact that $\displaystyle a = e^{ln(a)}$ and set $\displaystyle a=\frac{m}{m+k}$. All in all, I just feel like I'm getting even more jumbled up!

The book says that the limit for the original equation (up at the top) should be $\displaystyle e^{-k^2}$. Can anyone help me out?