let f(x)=(x^2+2x-15)/(3x^2+12x-15)

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

To find the horizontal and vertical asymptotes of a function, we need to analyze its behavior as x approaches infinity or negative infinity.

1. Horizontal Asymptotes:
To find the horizontal asymptotes, we need to compare the degrees of the numerator and denominator polynomials.

a) If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients. In this case, the degree of both the numerator and denominator is 2. So, the horizontal asymptote is the ratio of the leading coefficients, which are 1 (coefficient of x^2 in the numerator) and 3 (coefficient of x^2 in the denominator). Therefore, the horizontal asymptote is y = 1/3.

b) If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.

Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote of f(x) is y = 0.

2. Vertical Asymptotes:
To find the vertical asymptotes, we need to identify the values of x for which the denominator of the rational function goes to zero.

Set the denominator equal to zero and solve for x:

3x^2 + 12x - 15 = 0

This quadratic equation can be factored as:

(3x - 3)(x + 5) = 0

Setting each factor equal to zero, we find:

3x - 3 = 0 => x = 1
x + 5 = 0 => x = -5

Therefore, there are two vertical asymptotes: x = 1 and x = -5.

In summary,
- The horizontal asymptote of f(x) is y = 0.
- The vertical asymptotes of f(x) are x = 1 and x = -5.