How do you determine the vertical and horizontal asymptotes of the graph of each function # g(x) = (3x^2)/(x^2 - 9)#?

Question

Sadie Ackerman · Accepted Answer

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Serenity Jones · Answer

undefined To determine the vertical and horizontal asymptotes of the function $ g(x) = \frac{3x^2}{x^2 - 9} $:

1. Vertical asymptotes:
   Set the denominator equal to zero and solve for $ x $. Vertical asymptotes occur where the denominator equals zero, except where those values cancel with factors in the numerator. Here, the denominator is $ x^2 - 9 $, which factors into $ (x - 3)(x + 3) $. So, $ x $ cannot equal $ 3 $ or $ -3 $.

2. Horizontal asymptotes:
   If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $ y = 0 $. If the degrees are equal, divide the leading coefficients to find the horizontal asymptote. If the degree of the numerator is greater, there is no horizontal asymptote.

In this case, both the numerator and denominator have the same degree (2). Therefore, we divide the leading coefficients, which are both 3. So, the horizontal asymptote is at $ y = \frac{3}{1} = 3 $.