Find the horizontal and vertical asymptotes of f(x). Find the horizontal asympto

Question

Find the horizontal and vertical asymptotes of f(x). Find the horizontal asymptotes. Select the correct choice below and fill in any answer boxes within your choice. The horizontal asymptote(s) can be described by the line(s) (Simplify your answer. Type an equation. Use a comma to separate answers as needed. Use integers or fractions for any numbers in the equation.) There are no horizontal asymptotes. Find the vertical asymptotes. Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) can be described by the line(s) (Simplify your answer. Type an equation. Use a comma to separate answers as needed. Use integers or fractions for any numbers in the equation.) There are no vertical asymptotes.

Explanation / Answer

The horizontal asymptote, when the highest power of the numerator and denominator are the same, is simply the ratio of the associated coefficients. In this case, it is the coefficients for the x^2 terms, the highest power of the numerator and denominator. 3/1 = 3 The horizontal asymptote is y = 3 For the vertical asymptote, we first factor the numerator and denominator. 3x^2 + 5x - 2 = (3x-1)(x+2) x^2+2x-8 = (x+4)(x-2) Thus, the fraction is (3x-1)(x+2)/(x+4)(x-2) Note the absence of common terms in the numerator and denominator. Thus, the vertical asymptote equals the zeroes of (x+4)(x-2) The vertical asymptotes are x = -4 and x = 2

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