# How do you find the vertical, horizontal and slant asymptotes of: (7x-2 )/( x^2-3x-4)?

Jun 11, 2016

vertical asymptotes x = -1 , x = 4
horizontal asymptote y = 0

#### Explanation:

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation/s set the denominator equal to zero.

solve : ${x}^{2} - 3 x - 4 = 0 \Rightarrow \left(x - 4\right) \left(x + 1\right) = 0$

$\Rightarrow x = - 1 , x = 4 \text{ are the asymptotes}$

Horizontal asymptotes occur as

${\lim}_{x \to \pm \infty} , f \left(x\right) \to c \text{ (a constant)}$

divide terms on numerator/denominator by ${x}^{2}$

$\frac{\frac{7 x}{x} ^ 2 - \frac{2}{x} ^ 2}{{x}^{2} / {x}^{2} - \frac{3 x}{x} ^ 2 - \frac{4}{x} ^ 2} = \frac{\frac{7}{x} - \frac{2}{x} ^ 2}{1 - \frac{3}{x} - \frac{4}{x} ^ 2}$

as $x \to \pm \infty , f \left(x\right) \to \frac{0 - 0}{1 - 0 - 0}$

$\Rightarrow y = 0 \text{ is the asymptote}$

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (numerator-degree 1 , denominator-degree 2 ) Hence there are no slant asymptotes.
graph{(7x-2)/(x^2-3x-4) [-10, 10, -5, 5]}