Question

How do you find the vertical, horizontal or slant asymptotes for `f(x) = x/((x+3)(x-4))`?

Answer 1

Vertical Asymptotes are based on any factors in the Denominator while Horizontal/Slant Asymptote is based on the highest power of `x` in the Numerator and Denominator.

Explanation:

Vertical Asymptote:
Solve for `x` for each factor in the Denominator.
`x + 3 = 0` and `x - 4 = 0`
So Vertical Asymptote will appear at `x = -3` and `x = 4`

Horizontal Asymptote:
Let's say `f(x) = (ax^n)/(bx^m)`

If `n = m`, then Horizontal Asymptote is `y = a/b` (simplified).

If `n < m`, then Horizontal Asymptote is `y = 0`.

If `n > m`, then Horizontal Asymptote is None because it doesn't exist. Slant Asymptote only occurs if `n = m + 1`. We would have to use Long Division or Synthetic Division to find the Linear Slant Asymptote.

For the problem above, multiply the bottom factors:
`x/(x^2 -x - 12)`
Top Power of 1 `<` Bottom Power of 2.
So Horizontal Asymptote is `y = 0`