Searching Newton-Raphson failures!!

Question

I came up with a numerical method to find roots for nonlinear functions. Now I'm trying to see how better from other methods is. So.... where does Newton-Raphson fail? I came up with a function $f(x)=2{e}^{-x^{2}}-1$, that does not converge using N-R. The reason is that the derivative is very close to 0 value for most x. I think this happens to any function that asymptotically approcaches constant value. Others?