I have plotted function $(1+\frac{1}{x})^{x}$ using Maple and got following graph:
So it seems that function isn't defined on $(-1,0)$ interval , but if I take that $x=\frac{-1}{3}$ I can write:
$y=(1+\frac{1}{\frac{-1}{3}})^{\frac{-1}{3}}=(1-3)^{\frac{-1}{3}}=(-2)^{\frac{-1}{3}}=\frac{1}{(-2)^{\frac{1}{3}}}=\frac{1}{\sqrt[3]{-2}}=\frac{-1}{\sqrt[3]{2}}$
which is real number.So my question is :Is function $(1+\frac{1}{x})^{x}$ defined on $(-1,0)$ interval or not ?