Answer
The indeterminate form is $\pm\infty\cdot 0$
These limits equal $0.$
Work Step by Step
When $ x\rightarrow\infty$, the polynomial, depending on the sign of the leading term, veers off to either $+\infty$ or $-\infty$
For the exponential function,
$e^{-x} \qquad ( k^{Big\ negative }= Small)\qquad\rightarrow 0.$
The indeterminate form is $\pm\infty\cdot 0$.
Using a calculator, we have $e^{500}=1.4\times 10^{217}$, which is huge. A polynomial would have to have a degree of about 214 to catch up
The $e^{-x}$ approaches 0 at a much faster rate (orders of 10) than the polynomial approaches $\infty$, so the product approaches 0 as $ x\rightarrow\infty$.
