Derive asymptote algebraically?

I’m working my way through an Australian year 10 level mathematics text book and have been asked to give the asymptotes for y = 1/x.

The book has only mentioned asymptotes once before in passing and whist I can both “see” the asymptote by looking at the graph and have read that

If denominator′s degree > numerator′s degree, the horizontal asymptote is the x−axis: y=0.

I feel uneasy about merely memorising a rule, rote, without grasping a little of the proof behind it. Or am I over-reaching in my attempts to understand the material being presented at this stage?

Many of the explanations on the web I’ve come across seem to be dealing with the idea at the level of calculus and I’m wondering, is there an algebraic approach to deriving the asymptotes to this kind of rule?

The "usual" argument is going to involve explicit limit arguments. However, when I teach this to my precalculus classes, I try to avoid these explicit limit arguments, and handwave a little.

When $|x|$ is very large, the behaviour of a rational function will be "dominated" by the leading terms. The "right" way to demonstrate this is with limits, but we can build an intuition by noticing that if $m < n$, then it will eventually be the case that $$ |C x^m| < |x^n|. $$ In particular, if $|x| > \max\{ 1,|C| \}$, then $$ |C x^m| = |C| |x|^m < |x| |x|^m = |x|^{m+1} \le |x|^n.$$ What this shows is that every term in a polynomial is eventually smaller than the highest degree (leading) term, which means that the overall shape of a polynomial is determined by the leading term. A similar argument applies to rational functions. When teaching this, I will often write something like $$ \frac{a_n x^n + a_{n-1} x^{n-1} + \dotsb + a_1x + a_0}{b_m x^m + b_{m-1} x^{m-1} + \dotsb + b_1 x + b_0} = \frac{a_n x^n + \text{junk}}{b_m x^m + \text{junk}} \approx \frac{a_n}{b_m} \frac{x^n}{x^m} = C x^{n-m}. $$ The function $x \mapsto x^k$ is a "primitive function" in my instruction, which can have one of only a few possible graphs:

These "primitive functions" are then scaled (and possibly reflected across the $x$-axis by) $C$. Hence the asymptotic behaviour of a rational function is determined by the leading terms of the numerator and denominator. You get a horizontal asymptote at zero if the denominator has larger degree than the numerator; you get a horizontal asymptote at $y = a_n / b_m$ if the numerator and denominator have the same degree, and $a_n$ & $b_m$ are the leading coefficients of the numerator and denominator, respectively; and you get unbounded growth otherwise.

Note: The graphs above are only useful for understanding the behaviour of a rational function when you zoom way out. The function might look very different if you zoom in on it.