Question

Por favor ajuda numa questão de limites de um vestibular chamado Escola Naval, sem aplicar L'hopital? Obrigada.

lim (cotgx)^senx
x-->0+
a)inf
b)0
c) e
d)-1
e) 1

Answer 1

           $L=\lim\limits_{x\to0^+}(\cot x)^{\sin x}\\ \\ \\ L=\lim\limits_{x\to0^+}\left(\dfrac{\cos x}{\sin x}\right)^{\sin x}\\ \\ \\ L=\lim\limits_{x\to0^+}\left(\dfrac{1}{\sin x}\right)^{\sin x}\\ \\ \\ L=\lim\limits_{x\to0^+}\exp \left[\sin x \cdot \ln\left(\dfrac{1}{\sin x\right)}\right]\\ \\ \\ L=\exp\lim\limits_{x\to0^+} \dfrac{\ln\left(\dfrac{1}{\sin x\right)}}{\dfrac{1}{\sin x}}$
            
              $L=\exp\lim\limits_{y\to+\infty}\dfrac{\ln y}{y}\\ \\ \\ \text{Debe saber que: }1+\dfrac{1}{y}\leq \ln y\leq 1+y\;,\; \forall y\in \mathbb R^+\\ \\ \text{Por ende }\lim\limits_{y\to+\infty}\dfrac{\ln y}{y}=0 \text{ (apl\'iquese el teorema de Sandwich)}\\ \\ \\ \text{Finalmente deducimos que: }\\ \\ \boxed{L=1}$