We will prove that the facts below are equivalent.
- For a positive constant $a$, we have $\lim_{x\to \infty} a = a$, $\lim_{x\to \infty} \frac{a}{x} = 0$ and $\lim_{x\to \infty} ax = \infty$.
- For nonzero polynomials $p(x) = p_nx^n+\cdots p_1x+p_0$ and $q(x)= q_mx^m+\cdots q_1x+q_0\neq 0$ with leading coefficients $p_n> 0$ and $q_m> 0$, we have $$\lim_{x\to \infty} \frac{p(x)}{q(x)} = \begin{cases}\frac{p_n}{q_m}& n=m\\0 &m>n\\ \infty &n>m \end{cases}.$$
I am tired of having people think that (1) above is better than (2), and that (2) is a consequence of (1). I would rather we see them as equivalent.
To do this, we will need a definition of limits, as well as some relevant limit rules related to sums and constants (linearity rules). Generally people prove (1) first, and then use this to prove (2). However, if (2) is true, then 1 follows immediately by picking specific polynomials. So whichever of these two you use, they are equivalent (neither is truly better than the other). I think I can make that argument quite easily.
Suppose 1 is true. Then division by the smaller of $x^m$ in both $p$ and $q$ leads to the desired result, after applying linearity limit rules.
The polynomial rule clearly implies the first, by letting $p(x)=a$ and $q(x)=1$, $p(x)=a$ and $q(x)=x$, or $p(x)=ax$ and $q(x)=1$.
An argument can be made that proving 1, directly from the definition, is simpler. I would have to agree there. But that is because we focus on only way of expressing the definition. The limit definition is equivalent to lots of other things. Which way of expressing the limit definition is best? Depends on the context. For rational functions, learning rule 2 is much simpler than learning rule 1 with the associated linearity limit rules. Period. Context is everything.