What do subscript values indicate relative to polynomials? Fairly straight forward question, but in some cases I've identified some ambiguity. Consider the following function, F(x):
$a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$
and a second function, G(x):
$b_mx^m+b_{m-1}x^{m-1}+...+b_1x+b_0$
Straight away, it can be seen that this is the means of expressing a polynomial, and given this, the rational function $\frac{F(x)}{G(x)}$, can be assembled. Now consider the following rule when graphing such a function:


*

*If m = n, then $y=\frac{a_n}{b_m}$ is the horizontal asymptote


In this instance, what is meant by "m = n", because I always assumed subscript values refer to the position of a term in an expression, but if this were true, I don't understand how the horizontal asymptote can be inferred from this.
 A: As an add on to @dxiv comment, it can actually be shown that indeed if $a_n,b_m\neq0$, then $y=\frac {a_n}{b_m}$ is a horizontal asymptote to $P(x)=\frac {F(x)}{G(x)}$.
By definition, a horizontal asymptote exists if and only if $\lim_{x\to\pm\infty}P(x)=k$ for some constant$k\in\mathbb{R}$. 
Thus in general, we see:
$$\lim_{x\to\infty}P(x)=\lim_{x\to\infty}\frac{F(x)}{G(x)}=\lim_{x\to\infty}\frac{a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0}{b_mx^m+b_{m-1}x^{m-1}+...+b_1x+b_0}$$
Let's consider the case when $m=n$, then for $x\ne0$, we see:
$$\lim_{x\to\infty}P(x)=\lim_{x\to\infty}\frac{a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0}{b_nx^n+b_{n-1}x^{n-1}+...+b_1x+b_0}=\lim_{x\to\infty}\frac{a_n+\frac{a_{n-1}}{x}+...+\frac {a_1}{x^{n-1}}+\frac {a_0}{x^n}}{b_n+\frac{b_{n-1}}{x}+...+\frac {b_1}{x^{n-1}}+\frac {b_0}{x^n}}=\frac {a_n}{b_n}$$
So indeed, regarding your above question in the comments, the horizontal asymptote is:
$$\lim_{x\to\infty}\frac{F(x)}{G(x)}=\lim_{x\to\infty} \frac{4x^2+6x+8}{2x^2+4x+4}=\frac 42=2$$
Try considering what happens when $m>n$, or $m<n$.
