Question

How do you find the Vertical, Horizontal, and Oblique Asymptote given `(-7x + 5) / (x^2 + 8x -20)`?

Answer 1

Vertical asymptote: `x =-10` and `x = 2`.
Horizontal asymptote: `y = 0`.
Oblique (slant) asymptote: None.

Explanation:

To find the vertical asymptote, find all values of `x` that result in dividing by `0`. So factor the denominator:

`(-7x +5)/(x^2 +8x -20) rarr (-7x +5)/[(x +10)(x -2)]`

The values of `x = -10` and `x = 2` will result in dividing by zero, so those are the vertical asymptote.

To find the horizontal asymptote you look at the largest order of `x` in the numerator and compare it to the largest order of `x` in the denominator.

The numerator is first order and the denominator is second order. Since the order of the denominator is greater than the order of the numerator and the coefficient in front of `x^2` is one, the horizontal asymptote is `y = 0`

The oblique (slant) asymptote is only existent if the numerator is divisible by the denominator. Since the order of the denominator is larger than the numerator, the numerator is not divisible and there is no slant asymptote.