[Math] How to determine the end behavior of a rational function

Question

Example

$$\frac{6x + 2}{x^2 – 9} = \frac{6x + 2}{(x + 3)(x – 3)}$$

I know how to find the vertical and horizontal asypmtotes and everything, I just don't know how to find end behavior for a RATIONAL function without plugging in a bunch of numbers.

Answer 1

If you are concerned by the behavior of the function when $x$ starts to be large, just perform the long division of polynomials.

For $$f(x)=\frac{6x+2}{x^2-9}$$ this will give $$f(x)\approx \frac{6}{x}+\frac{2}{x^2}$$ and then the asymptote would be function $\frac{6}{x}$.

Changing to $$g(x)=\frac{6x^2+2}{x^2-9}$$ this will give $$g(x)\approx 6+\frac{56}{x^2}$$ and then the asymptote would be function $6$, an horizontal asymptote.

Changing to $$h(x)=\frac{6x^3+2}{x^2-9}$$ this will give $$h(x)\approx 6 x+\frac{54}{x}+\frac{2}{x^2}$$ and then the asymptote would be function $6 x$, an oblique asymptote.

You could notice that this simple division gives you the asymptote as well as the manner the function appoaches it.