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 \(\lim\limits_{x\rightarrow c}[ f(x) ] ^{g(x) }\).

Do not stop after finding \(\lim\limits_{x\rightarrow c} \ln\;{y} { (=L)}\). Remember, we want to find \(\lim\limits_{x\rightarrow c}{ y}(=e^{L}) .\)

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 \(\lim\limits_{x\rightarrow 0^{+}}\;\ln y=0\), \(\lim\limits_{x\rightarrow 0^{+}}y=e^{0}=1\).