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

1 Answer
KillerBunny
Oct 21, 2015

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.

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