Given the following rational function, find

a. the horizontal asymptote(s),
b. the vertical asymptote(s), if any, and
c. the oblique asymptote(s), if any.

f(x)=(x^2-x-2)/(2x^2-x-10)

To find the horizontal asymptote(s) of the given rational function, let's examine the degrees of the numerator and denominator.

a. Horizontal Asymptote(s):
The degrees of the numerator and denominator are both 2. To determine the horizontal asymptote(s), compare the leading terms of both polynomials:

Leading term of the numerator: x^2
Leading term of the denominator: 2x^2

Since the degrees of the leading terms are the same, we compare the coefficients:

Coefficient of the numerator: 1
Coefficient of the denominator: 2

If the coefficient of the numerator is less than the coefficient of the denominator, then the horizontal asymptote is at y = 0. However, if the degree of the numerator is equal to the degree of the denominator and the coefficients are the same, then the horizontal asymptote is at y = coefficient of the numerator / coefficient of the denominator. In this case, since 1 is less than 2, the horizontal asymptote is at y = 0.

Therefore, the function has a horizontal asymptote at y = 0.

b. Vertical Asymptote(s):
To find the vertical asymptote(s), set the denominator equal to 0 and solve for x. In this case, the denominator is 2x^2 - x - 10.

Factorizing the denominator, we have:
2x^2 - x - 10 = (2x + 5)(x - 2)

Setting each factor equal to zero:
2x + 5 = 0 --> 2x = -5 --> x = -5/2
x - 2 = 0 --> x = 2

Thus, the vertical asymptotes occur at x = -5/2 and x = 2.

c. Oblique Asymptote(s):
To find the oblique asymptote(s), divide the numerator by the denominator using long division or synthetic division. In this case, the numerator is x^2 - x - 2 and the denominator is 2x^2 - x - 10.

Performing the division, we get:
1/2

___________
2x^2 - x - 10) | x^2 - x - 2
- (x^2 - (x - 2)
______________________
x + 0

The remainder of the division is x + 0, which simplifies to just x. Therefore, the oblique asymptote is y = 1/2x.

To summarize:
a. The function has a horizontal asymptote at y = 0.
b. The function has vertical asymptotes at x = -5/2 and x = 2.
c. The function has an oblique asymptote at y = 1/2x.