Práctica 3 ejercicio 9o

EJERCICIO 9: ...

(o) $ \big(n^4 +1 \big)^{1/2n}$

Solución:
$\lim\limits_{n\to \infty} \big(n^4 +1 \big)^{1/2n} = \lim\limits_{n\to \infty} \left(n^4 \left(1+\dfrac{1}{n^4}\right) \right)^{1/2n} = \lim\limits_{n\to \infty} \left(n^4 \right)^{1/2n} \left(1+\dfrac{1}{n^4}\right) ^{1/2n} = $

$\lim\limits_{n\to \infty} \Big(\left(n^4 \right)^{1/2}\Big)^{1/n} \left(1+\dfrac{1}{n^4}\right) ^{1/2n} =
\lim\limits_{n\to \infty} \Big(n^2\Big)^{1/n} \left(1+\dfrac{1}{n^4}\right) ^{1/2n} =$

$\lim\limits_{n\to \infty} (\underbrace{\sqrt[n]{n}}_{\to 1})^{2} \underbrace{\left(1+\dfrac{1}{n^4}\right) ^{\overbrace{1/2n}^{\to 0}}}_{\to 1} = 1$