# How do you find the asymptotes for f(x)=(3x^2+2) / (x^2 -1)?

Jun 8, 2016

vertical asymptotes x = ± 1
horizontal asymptote y = 3

#### 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^2-1=0rArrx^2=1rArrx=±1" 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 ${x}^{2}$

$\frac{\frac{3 {x}^{2}}{x} ^ 2 + \frac{2}{x} ^ 2}{{x}^{2} / \left({x}^{2}\right) - \frac{1}{x} ^ 2} = \frac{3 + \frac{2}{x} ^ 2}{1 - \frac{1}{x} ^ 2}$

as $x \to \pm \infty , f \left(x\right) \to \frac{3 + 0}{1 - 0}$

$\Rightarrow y = 3 \text{ is the asymptote}$
graph{(3x^2+2)/(x^2-1) [-10, 10, -5, 5]}