Are you allowed to use l'Hopital's Rule? NB: This limit will only be valid if it's a right hand limit: $\displaystyle \mathop {\lim }\limits_{x \to 0^{\color{red}+}} \left( {x^x } \right)$.
Let y = x^x. Then ln(y) = x*ln(x) = ln(x)/(1/x). You can apply l'Hospital's Rule to this, and get the limit, as x tends to zero from the right, of ln(y). Then raise this value as a power on e, to get the limit of y.