Question

How do you find the limit `lima^x` as `x->oo`?

Answer 1

Use `a^x = e^(xlna)` and consider cases.

Explanation:

For `0 < a < 1`, we have `lna < 0`, so

as `xrarroo`, we get `xlnararr-oo` and so `a^x rarr 0`.

For `a=0`, we get the indeterminate form `1^oo`.

For `a > 1`, we have `lna > 0`, so

as `xrarroo`, we get `xlnararroo` and so `a^x rarr oo`.