Question

O valor do limite lim x^x é (x-->0+) x tende a zero pela direita

Answer 1

$\lim\limits_{x\to 0^+}x^x=\exp\left(\lim\limits_{x\to0^+}x\ln x\right)\\ \\ \lim\limits_{x\to 0^+}x^x=\exp\left(\lim\limits_{x\to0^+}\dfrac{\ln x}{\frac{1}{x}}\right)\\ \\ \text{Regra de L'Hospital: }\\ \\ \lim\limits_{x\to 0^+}x^x=\exp\left(\lim\limits_{x\to0^+}\dfrac{(\ln x)'}{(\frac{1}{x})'}\right)\\ \\ \lim\limits_{x\to 0^+}x^x=\exp\left(\lim\limits_{x\to0^+}\dfrac{\frac{1}{x}}{-\frac{1}{x^2}}\right)$

$\lim\limits_{x\to 0^+}x^x=\exp\left(\lim\limits_{x\to0^+}-x\right)\\ \\ \lim\limits_{x\to 0^+}x^x=\exp 0\\ \\ \\ \boxed{\lim\limits_{x\to 0^+}x^x=1}$