Question

resuelvan este limite

Answer 1

$L=\lim\limits_{x\to\infty}\left(1+\dfrac{x}{6}\right)^{\dfrac{\frac{1}{2}}{\frac{x}{6}}}\\ \\ \\ \texttt{Sea }y=\dfrac{x}{6}\texttt{ si }x\to\infty\texttt{ entonces }y\to \infty\texttt{ por ende: }\\ \\ \\ L=\lim\limits_{y\to\infty}\left(1+y\right)^{\dfrac{\frac{1}{2}}{y}}\\ \\ \\ L=\sqrt{\lim\limits_{y\to\infty}\left(1+y\right)^{\dfrac{1}{y}}}\\ \\ \\ (0\leq L)~~L^2=\lim\limits_{y\to\infty}\left(1+y\right)^{\dfrac{1}{y}}$


$\mathcal{L}=\lim\limits_{y\to\infty}\left(1+y\right)^{\dfrac{1}{y}}\\ \\ \\ \mathcal{L}=\lim\limits_{y\to\infty}\exp\left[\dfrac{\ln(1+y)}{y}\right]\\ \\ \\ \mathcal{L}=\exp 0\\ \\ \mathcal{L}=1\\ \\ (0\leq L)~~L^2=1\\ \\ \boxed{L=1}$