I'm studying asymptotes and wondering if there's a quicker way to determine where the graph is going.

Let me explain with an example:

$$f(x)=\frac{2x^3+x^2-1}{x^2-1}$$

So there are two vertical asymptotes x=-1 and x=1

To know how the curve is turning i check f(1.001), f(-1.001), f(-0.999), f(0.999).

Doing it by hand is pretty slow, is there a better way of knowing where the function goes when x approaches the asymptotes?