# What is the limit of [x^2+3x+4]/[x^3-3x^2+1] + [x^4-x]/[x^5] as x goes to infinity?

It is $0$
Both ratios go to $0$ as $x$ increases without bound, so the sum also goes to $0$.
For $\frac{{x}^{2} + 3 x + 4}{{x}^{3} - 3 {x}^{2} + 1}$, as $x$ increases without bound, the numerator and denominator race to infinity. Because the denominator has greater degree, it wins the race (it gets big faster than the numerator). So the ratio goes to $0$.
The ratio $\frac{{x}^{4} - x}{{x}^{5}}$ is similar.