Question

Which of the following rational functions has a horizodtal asymptote at y=3 and vertical asymptotes at x=5 and x=-2 ? y=(x^(2))/(x^(2)-3x-10) y=(3x^(2))/(x^(2)-3x-10) y=(x^(2))/(x^(2)+3x-10) y=(3x^(2))/(x^(2)+3x-10)

Answer 1

To determine the rational function with specific asymptotes, analyze the behavior at those values. The function with a horizontal asymptote at y=3 and vertical asymptotes at x=5 and x=-2 is y=(3x^(2))/(x^(2)-3x-10).

Step-by-step explanation:

To determine which of the given rational functions has a horizontal asymptote at y=3 and vertical asymptotes at x=5 and x=-2, we need to analyze the behavior of the functions at these values.

For the given options, the correct function is y=(3x^(2))/(x^(2)-3x-10). At x=5 and x=-2, the denominator becomes zero, which results in vertical asymptotes. And as x approaches positive or negative infinity, the ratio of the leading terms of the numerator and denominator becomes 3, leading to a horizontal asymptote at y=3.