# Lim, sup, and inf

Calculus Level 2

Let $x_n$ be the sequence

$2,\ -2^{1/2},\ 2^{1/3},\ -2^{1/4},\ 2^{1/5},\ -2^{1/6},\ \ldots.$

What are the values of

$\inf x_n, \ \ \liminf_{n\to\infty} x_n, \ \ \lim_{n\to\infty} x_n, \ \ \limsup_{n\to\infty} x_n, \ \ \sup x_n \, ?$

Notation: In the choices, $\text{DNE}$ means "does not exist."

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