The general solution of the homogeneous equation $Ly = 0$ is given by:
$y(t) = c_1(t)e^{λ_1t} + c_2(t)e^{λ_2t} ... c_m(t)e^{λ_mt}$, where $c_j(t)$ is an arbitrary polynomial of degree $k_j − 1$.
If $r$ is a nonnegative integer and $\lambda \in \mathbb{C}$, how can I show that $\lim_{t\to\infty} |(t^r)(e^{λt})| = 0$ if $\Re(λ) < 0$?
Hint
$$\left|e^{z}\right| = e^{\text{Re}(z)}$$