Answer
The indeterminate form is $\displaystyle \frac{\pm\infty}{\infty}$.
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, $e^{x}\rightarrow\infty.$
The indeterminate form is $\displaystyle \frac{\pm\infty}{\infty}$.
Using a calculator, $e^{500}=1.4\times 10^{217}$, which is huge. A polynomial would have to have a degree of about 214 to catch up. The denominator increases at a faster rate (orders of 10) than the polynomial in the numerator, so the ratio approaches 0 as $ x\rightarrow\infty$.
