Question

Find the vertical, horizontal, and oblique asymptotes, if any, for f(x)=(3x³ + 20x² + 14x - 65)/(x² + 8x +15)

A. Vertical: x=5, x=−3, Horizontal: y=0, Slant: y=3x+4

B. Vertical:x=−5, x=−3, Horizontal: y=0, Slant: y=3x−4

C. Vertical: x=−5, x=3, Horizontal: y=0, Slant: y=3x−4

D. Vertical: x=5, x=−3, Slant: y=3x+4

Answer 1

The vertical asymptotes are x = -3 and x = -5. There is no horizontal asymptote. The slant asymptote is y = 3x + 4. Hence the correct answer is option D

Step-by-step explanation:

To find the vertical, horizontal, and oblique asymptotes of the function f(x)=(3x³ + 20x² + 14x - 65)/(x² + 8x +15), we need to analyze the behavior of the function as x approaches positive infinity, negative infinity, and the values that make the denominator equal to zero.

- Vertical asymptotes: Set the denominator equal to zero and solve for x. In this case, the denominator factors to (x+3)(x+5). Therefore, the vertical asymptotes are x = -3 and x = -5.

- Horizontal asymptote: The degree of the numerator is 3 and the degree of the denominator is 2. Since the degree of the numerator is greater, there is no horizontal asymptote.

- Slant asymptote: Since the degree of the numerator is one greater than the degree of the denominator, there is a slant asymptote. To find it, perform long division between the numerator and denominator. The result is y = 3x + 4.

Hence the correct answer is option D