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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#.

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 indicates that this horizontal line serves as your function:

graph{(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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Answer 2

Isaiah Anthony

The vertical asymptotes are x = -3 and x = 3. There are no horizontal asymptotes.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.