Question

lim ( sec x ) ^(1/x²)
x->0

Alguém pode me ajudar?

Answer 1

$L=\lim\limits_{x\to 0}(\sec x )^{\frac{1}{x^2}}\\ \\ \\ \ln L=\ln\lim\limits_{x\to 0}(\sec x )^{\frac{1}{x^2}}\\ \\ \\ \ln L=\lim\limits_{x\to 0}\ln(\sec x )^{\frac{1}{x^2}}\\ \\ \\ \ln L=\lim\limits_{x\to 0}\dfrac{\ln \sec x}{x^2}\\ \\ \\ \ln L=\lim\limits_{x\to 0}-\dfrac{\ln \cos x}{x^2}\\ \\ \\ \texttt{L'Hospital}\\ \\ \\ \ln L=\lim\limits_{x\to 0}-\dfrac{\frac{-\sin x}{\cos x}}{2x}\\ \\ \\ \ln L=\dfrac{1}{2}\lim\limits_{x\to 0} \dfrac{\sin x}{x}\cdot \sec x\\ \\ \\ \ln L= \dfrac{1}{2}$


$\huge\boxed{L=\sqrt{e}}$