Lim, sup, and inf

Calculus Level 3

Let xn x_n be the sequence

2, 21/2, 21/3, 21/4, 21/5, 21/6, .2,\ -2^{1/2},\ 2^{1/3},\ -2^{1/4},\ 2^{1/5},\ -2^{1/6},\ \ldots.

What are the values of

infxn,  lim infnxn,  limnxn,  lim supnxn,  supxn?\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, DNE\text{DNE} means "does not exist."

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