Lim, sup, and inf

Calculus Level 3

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 \[ \inf x_n, \quad \liminf_{n\to\infty} x_n, \quad \lim_{n\to\infty} x_n, \quad \limsup_{n\to\infty} x_n, \quad \sup x_n \; ? \]

Notation:

  • In the choices, \(\text{DNE} \) means "does not exist."
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