$\lim_{x\to\infty}\dfrac{P(x)}{Q(x)} = \lim_{x\to-\infty}\dfrac{P(x)}{Q(x)} = L$; $P(x) Q(x)$ - polynomials with degree $P(x) \leq $ degree $Q(x)$

Question

This is a true statement, right? And $L = 0$. I could prove this using algebraic limit theorems and taking $P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ and $Q(x) = b_m x^m + b_{m-1} x^{m-1} + ... + b_0$.

$\dfrac{P(x)}{Q(x)} = \dfrac{a_n x^{n-m} + a_{n-1} x^{n-m-1} + \dots + a_0 x^{-m} } {b_m + b_{m-1} x^{-1} + \dots + b_0 x^{-m}}$

Here, everything goes to $0$ at the numerator thus $L = 0$.

Is this valid argument?