PART 11：例題-不定型 \({0^0}\)

求極限 \(\lim\limits_{x \to {0^ + }} {x^x}\)
SOL :

要利用性質 \(f({f^{ - 1}}(x)) = x\) ， \(f\) 為自然指數， \({f^{ - 1}}\) 為自然對數，

將 \({x^x}\) 取對數，再取指數可恢復原狀，

也就是 \({e^{\ln ({x^x})}} = {x^x}\)\(\Rightarrow {x^x} = {e^{\ln ({x^x})}} = {e^{x\ln x}} = {e^{\frac{{\ln x}}{x}}}\)

原題目 \(\lim\limits_{x \to {0^ + }} {x^x}\) = \(\lim\limits_{x \to {0^ + }} {e^{\frac{{\ln x}}{x}}}\)  \( = {e^{\lim\limits_{x \to {0^ + }} \frac{{\ln x}}{x}}}\)  \( = {e^{\;\lim\limits_{x \to {0^ + }} \frac{{\ln x}}{x}}}\)  \( = {e^{\;0}} = 1\)

(利用前面例題的結果)