How do you find the vertical, horizontal or slant asymptotes for #f(x) = (3x^2 - 3x- 36) / (2x^2 + 9x +4) #?

Question

Lily Anderson · Accepted Answer

vertical asymptotes #x=-4,x=-1/2#
horizontal asymptote #y=3/2# The denominator of f(x) cannot be zero as this is undefined. Equating the denominator to zero and solving gives the values that x cannot be and if the numerator is non-zero for these values then they are asymptotes. solve: #2x^2+9x+4)=0rArr(2x+1)(x+4)=0# #rArrx=-4" and " x=-1/2" are the asymptotes"# Horizontal asymptotes occur as #lim_(xto+-oo),f(x)toc" (a constant)"# divide terms on numerator/denominator by the highest power of x, that is #x^2# #((3x^2)/x^2-(3x)/x^2-36/x^2)/((2x^2)/x^2+(9x)/x^2+4/x^2)=(3-3/x-36/x^2)/(2+9/x+4/x^2)# as #xto+-oo,f(x)to(3-0-0)/(2+0+0)# #rArry=3/2" 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{(3x^2-3x-36)/(2x^2+9x+4) [-20, 20, -10, 10]}

Cora Alexander · Answer

undefined To find the vertical asymptotes of the function $ f(x) = \frac{3x^2 - 3x - 36}{2x^2 + 9x + 4} $, set the denominator equal to zero and solve for $ x $. The values of $ x $ obtained are the vertical asymptotes.

To find the horizontal asymptotes, compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $ y = 0 $. If the degrees are equal, divide the leading coefficients of the numerator and denominator to find the horizontal asymptote. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

To find the slant asymptote, perform polynomial long division or synthetic division to divide the numerator by the denominator. The quotient obtained represents the equation of the slant asymptote. If the degrees of the numerator and denominator are equal, there is no slant asymptote.

For the given function:
1. Vertical asymptotes: Solve $ 2x^2 + 9x + 4 = 0 $ to find the vertical asymptotes.
2. Horizontal asymptote: Compare degrees. If needed, divide the leading coefficients.
3. Slant asymptote: Perform polynomial long division or synthetic division to find the quotient, representing the slant asymptote equation.