quality:corruption
Rational functions

For n a nonnegative integer and constants a 1, a 2, a 3, …, a n {\displaystyle a_{1},a_{2},a_{3},\ldots,a_{n}} and b 1, b 2, b 3, …, b n, {\displaystyle b_{1},b_{2},b_{3},\ldots,b_{n},}

lim x → ∞ a 1 x n + a 2 x n − 1 + a 3 x n − 2 + ⋯ + a n b 1 x n + b 2 x n − 1 + b 3 x n − 2 + ⋯ + b n = a 1 b 1 {\displaystyle \lim _{x\to \infty }{\frac {a_{1}x^{n}+a_{2}x^{n-1}+a_{3}x^{n-2}+\dots +a_{n}}{b_{1}x^{n}+b_{2}x^{n-1}+b_{3}x^{n-2}+\dots +b_{n}}}={\frac {a_{1}}{b_{1}}}}

This can be proven by dividing both the numerator and denominator by x. If the numerator is a polynomial of higher degree, the limit does not exist. If the denominator is of higher degree, the limit is 0.