Question

What are the vertical and horizontal asymptotes of `y = ((x-3)(x+3))/(x^2-9)`?

Answer 1

The function is a constant line, so its only asymptote are horizontal, and they are the line itself, i.e. `y=1`.

Explanation:

Unless you misspelled something, this was a tricky exercise: expanding the numerator, you get `(x-3)(x+3)=x^2-9`, and so the function is identically equal to `1`.

This means that your function is this horizontal line:

graph{((x-3)(x+3))/(x^2-9) [-20.56, 19.99, -11.12, 9.15]}

As every line, it is defined for every real number `x`, and so it has no vertical asymptotes. And in a sense, the line is its own vertical asymptote, since

`lim_{x\to\pm\infty} f(x)=lim_{x\to\pm\infty} 1=1`.