Because you're interested in the behaviour of the function as $|x|$ grows large. You can split the function up. I'll take an example:
$$\frac{3x^2+1}{x+2}$$ $$\frac{3x(x+2)-6x+1}{x+2}$$ $$3x+\frac{-6x+1}{x+2}$$ $$3x+\frac{-6(x+2)+13}{x+2}$$ $$3x-6+\frac{13}{x+2}$$
at each step I'm rewriting the highest term in a way that involves $x+2$, and then adjusting the lower terms so that the expansion comes out right. Now look what happens to the fraction as $x$ grows large:
$$\frac{13}{102},\frac{13}{1002},\frac{13}{10002}, \cdots$$
It gets smaller and smaller and goes to $0$. You can show the same thing for the other direction as well (negative numbers that get closer and closer to zero). So an asymptote is concerned with behaviour _as $x$ grows large_ , and as $x$ grows large the fractional portion drops out to $0$. It only influences the function near the origin, which the asymptote isn't concerned with.