Find $\lim\limits_{x\rightarrow 0^{+}}x^{x}$

Solution The expression $x^{x}$ is an indeterminate form at $0^{+}$ of the type $0^{0}.$ We follow the steps for finding $\lim\limits_{x\rightarrow c}[ f(x) ] ^{g(x) }$.

Step 1 Let $y=x^{x}.$ Then $\ \ln y=x\ln x.$

Step 2 $\lim\limits_{x\rightarrow 0^{+}}\;\ln y=\lim\limits_{x\rightarrow 0^{+}}( x\;\ln x) =0$ [from Example 7(a)].

Step 3 Since $\lim\limits_{x\rightarrow 0^{+}}\;\ln y=0$, $\lim\limits_{x\rightarrow 0^{+}}y=e^{0}=1$.