Fixing $t > 0$, consider that the statement:
$e^{-tx} \le \frac{1}{t^2x^5}$
is true for $\forall x$ large enough no matter what $t$ happens to be (this due to the nature of the exponential function getting exponentially smaller so at some point "taking over" the smallness of $\frac{1}{t^2x^5}$). Yet this will only be true for $\forall x \ge x_c$ that will depend on $t$.
My question is whether or not can we place a bound on $x_c$ so that no matter what $t > 0$ we choose, the inequality expressed above will hold? That is, can we find some $x_c$ s.t. $0 \le x_c$ whereby $\forall x \ge x_c$ we have that $\forall t > 0$, $e^{-tx} \le \frac{1}{t^2x^5}$?