$\frac{c^t e^t}{t^{t+1/2}}$ and $e^{-kt^2}$ which decay faster as $t \rightarrow \infty$？ where $c$ is constant

Question

$\frac{c^t e^t}{t^{t+1/2}}$ and $e^{-kt^2}$ which decay faster as $t \rightarrow \infty$？ where $c,k$ are constants . how to see that?

Answer 1

Consider the respective logarithms $t(\ln c+1-\ln t)-\frac{1}{2}\ln t\sim -t\ln t,\,-kt^2$. The latter heads to $-\infty$ faster as $t\to\infty$, so $e^{-kt^2}$ is the faster-decaying function.