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

May 7, 2018

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 $f \left(x\right) = \frac{a {x}^{n}}{b {x}^{m}}$

If $n = m$, then Horizontal Asymptote is $y = \frac{a}{b}$ (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:
$\frac{x}{{x}^{2} - x - 12}$
Top Power of 1 $<$ Bottom Power of 2.
So Horizontal Asymptote is $y = 0$