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

Dec 31, 2016

Use ${a}^{x} = {e}^{x \ln a}$ and consider cases.

Explanation:

For $0 < a < 1$, we have $\ln a < 0$, so

as $x \rightarrow \infty$, we get $x \ln a \rightarrow - \infty$ and so ${a}^{x} \rightarrow 0$.

For $a = 0$, we get the indeterminate form ${1}^{\infty}$.

For $a > 1$, we have $\ln a > 0$, so

as $x \rightarrow \infty$, we get $x \ln a \rightarrow \infty$ and so ${a}^{x} \rightarrow \infty$.