A function $f(x)=\frac{P(x)}{Q(x)}$ is
1) improper if degree of $P$ $\ge$ degree of $Q$.
2) proper if degree of $P$ $\lt$ degree of $Q$.
I get this idea, but why do we identify these functions as proper or improper? Are there any characteristics or things to be careful about when dealing with them? My Calculus teacher mentioned this concept while explaining how to draw a graph of the function $y=\frac{2x^2}{x^2-1}$ which is improper, apparently. Does the graph of an improper function vary from that of a proper function?
We talk of proper or improper rational functions since any improper rational function can be simplified to get a proper rational function by long division, that is for example
$$y=\frac{2x^2}{x^2-1}=2\frac{x^2-1+1}{x^2-1}=2\frac{x^2-1}{x^2-1}+2\frac{1}{x^2-1}=2+\frac{2}{x^2-1}$$
Therefore the graph for $\frac{2x^2}{x^2-1}$ can be obtained by a vertical positive translation of 2 for the function $\frac{2}{x^2-1}$ wich has horizontal asympthotes for $x\to \pm \infty$ and vertical asympthotes for $x=\pm 1$.