Answer
$a>0$, $k>0$
Work Step by Step
When $t\rightarrow \infty$, $e^{kt}\rightarrow \infty$ when $k>0$ and $e^{kt}\rightarrow 0$ when $k<0$. As $a$ is a constant, to have $ae^{kt}\rightarrow \infty$, we need to have $e^{kt}\rightarrow \infty$, so $k>0$.
Because $e^{kt}\rightarrow \infty$ when $t\rightarrow \infty$, to obtain $ae^{kt}\rightarrow\infty$, we must have $a>0$.
So the values of $a$ and $k$ must be both positive so that:
$$
\lim _{t \rightarrow \infty} a e^{k t}=\infty .
$$
