How do you find the vertical, horizontal or slant asymptotes for # r(x)= ((2x^2+14x-36)/(x^2+x-12))#?

Question

Serenity Jones · Accepted Answer

vertical asymptotes x = -4 , x = 3
horizontal asymptote y = 2

The denominator of r(x) cannot equal zero as this is undefined. Setting the denominator equal to zero and solving gives the values that x cannot be and if the numerator is non-zero for these values then they are vertical asymptotes. solve: #x^2+x-12=0rArr(x+4)(x-3)=0# #rArrx=-4,x=3" are the asymptotes"# Horizontal asymptotes occur as #lim_(xto+-oo),r(x)toc" (a constant)"# divide terms on numerator/denominator by the highest exponent of x , that is #x^2# #((2x^2)/x^2+(14x)/x^2-36/x^2)/(x^2/x^2+x/x^2-12/x^2)=(2+14/x-36/x^2)/(1+1/x-12/x^2)# as #xto+-oo,r(x)to(2+0-0)/(1+0-0)# #rArry=2" is the asymptote"# Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (both of degree 2) Hence there are no slant asymptotes. graph{(2x^2+14x-36)/(x^2+x-12) [-20, 20, -10, 10]}

Elias Ackerman · Answer

undefined To find the vertical, horizontal, or slant asymptotes of the function $r(x) = \frac{2x^2 + 14x - 36}{x^2 + x - 12}$, follow these steps:

1. **Vertical Asymptotes**: Vertical asymptotes occur where the denominator of the rational function equals zero and the numerator doesn't. Set the denominator $x^2 + x - 12$ equal to zero and solve for $x$. These values of $x$ will give the vertical asymptotes.

2. **Horizontal Asymptotes**: Horizontal asymptotes can be found by examining the behavior of the function as $x$ approaches positive or negative infinity. If the degrees of the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

3. **Slant Asymptotes**: Slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator. To find a slant asymptote, perform polynomial long division or synthetic division to divide the numerator by the denominator. The quotient obtained will represent the equation of the slant asymptote.

Perform these steps to find the vertical, horizontal, or slant asymptotes for the given function $r(x)$.