Evaluating $\lim_{x\to 0} (1-x)^{\frac 1x}$ How do you evaluate this limit?
$$\lim_{x\to 0} (1-x)^{\frac 1x}$$

I looked at it graphically (here is the graph), but it doesn't look like the limit is any nice number. Does anyone know how to find this?
One of the definitions of $e^y$ is $\lim_{n \to \infty}\left(1+\dfrac{y}n\right)^n$. In your case, replace $x$ by $1/n$, we then have
$$\lim_{x \to 0}\left(1-x\right)^{1/x} = \lim_{n \to \infty}\left(1-\dfrac1n\right)^n = e^{-1}$$
I'm pretty sure the limit is $e^{-1}$.  In fact $(1+\frac1n)^n\rightarrow e$.