What are the horizontal and vertical asymptotes of the following equations?

Question

a.) r(x) = (x - 5)(x + 3)

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(8x + 1)(3x - 5)

b.) r(x) = 5x^2 + 7x

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x^4 - 1

Answer 1

Vertical asymptotes will be at the values of x that make the denominator zero and horizontal asymptotes will be the limit of the function as x approaches ±∞. For the latter, look at the highest power expression in the numerator and denominator. If the numerator has a higher power than the denominator, then there are no horizontal asymptotes. If the denominator has a higher power than the numerator, then the horizontal asymptote is at y=0. If the highest powers are the same, then the horizontal asymptote will be the ratio of the coefficients of those two expressions.

a). The vertical asymptotes are at x=5 and x=-3 because these values make the denominator zero. The horizontal asymptote is at y=1/24 because both the numerator and denominator's highest power term is x² and the ratio of the coefficients of those terms is 1/24.

b). We first need to factor the denominator to get the vertical asymptotes:

r(x) = (5x² + 7x)/((x²+1)(x-1)(x+1))

So the vertical asymptotes are at x=1 and x=-1. The horizontal asymptote is at y=0 because the denominator has a higher power than the numerator (x⁴ compared to x²).