# How do you find the vertical, horizontal or slant asymptotes for (x² + x - 2) /( x² + 4x + 3)?

Apr 18, 2016

vertical asymptotes x = -3 , x = -1
horizontal asymptote y = 1

#### 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 + 4x + 3 = 0 → (x+3)(x+1) = 0

$\Rightarrow x = - 3 , x = - 1 \text{ are the asymptotes }$

Horizontal asymptotes occur as ${\lim}_{x \to \pm \infty} f \left(x\right) \to 0$

If the degree of the numerator and denominator are equal , as in this case , then the equation is the ratio of leading coefficients, which are the coefficients of the ${x}^{2} \text{ terms }$

$\Rightarrow y = \frac{1}{1} = 1 \text{ is the asymptote }$

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