Question

Explain how to find the vertical and horizontal asymptotes of a rational function. Then find the asymptotes of the rational function (2x + 2)/(x^2 - 1).

a) Vertical asymptote: x = -1, Horizontal asymptote: y = 0
b) Vertical asymptote: x = 1, Horizontal asymptote: y = 2
c) Vertical asymptote: x = -1, Horizontal asymptote: y = 2
d) Vertical asymptote: x = 1, Horizontal asymptote: y = 0

Answer 1

To find the vertical and horizontal asymptotes of a rational function, analyze the behavior as x approaches infinity and as x approaches values that make the denominator zero. For the given function, x = -1 is the vertical asymptote and there is no horizontal asymptote.

Step-by-step explanation:

To find the vertical and horizontal asymptotes of a rational function, we need to analyze the behavior of the function as x approaches positive and negative infinity, and as x approaches the values that make the denominator zero.

1. Vertical asymptotes occur when the denominator of the rational function equals zero. In this case, x^2 - 1 = 0 gives us x = 1 and x = -1. So, the vertical asymptotes of the given function are x = 1 and x = -1.
2. Horizontal asymptotes occur when the degree of the numerator is less than or equal to the degree of the denominator. In this case, the degrees are both 1. So, we don't have a horizontal asymptote.

Therefore, the correct answer is a) Vertical asymptote: x = -1, Horizontal asymptote: y = 0.