Question

What is the limit of `x^n`?

Answer 1

`lim_(n->oo) x^n` behaves in seven different ways according to the value of `x`

Explanation:

If `x in (-oo, -1)` then as `n->oo`, `abs(x^n)->oo` monotonically, but alternates between positive and negative values. `x^n` does not have a limit as `n->oo`.

If `x = -1` then as `n->oo`, `x^n` alternates between `+-1`. So again, `x^n` does not have a limit as `n->oo`.

If `x in (-1, 0)` then `lim_(n->oo) x^n = 0`. The value of `x^n` alternates between positive and negative values but `abs(x^n) -> 0` is monotonically decreasing.

If `x = 0` then `lim_(n->oo) x^n = 0`. The value of `x^n` is constant `0` (at least for `n > 0`).

If `x in (0, 1)` then `lim_(n->oo) x^n = 0` The value of `x^n` is positive and `x^n -> 0` monotonically as `n->oo`.

If `x = 1` then `lim_(n->oo) x^n = 1`. The value of `x^n` is constant `1`.

If `x in (1, oo)` then as `n->oo`, then `x^n` is positive and `x^n->oo` monotonically. `x^n` does not have a limit as `n->oo`.