How do you find the Vertical, Horizontal, and Oblique Asymptote given #f(x)=(5x-15) /( 2x+4)#?

Question

Sadie Alexander · Accepted Answer

vertical asymptote x = -2
horizontal asymptote #y=5/2# The denominator of f(x) cannot be zero as this is undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote. solve: 2x + 4 = 0 → x = -2 is the asymptote Horizontal asymptotes occur as #lim_(xto+-oo),f(x)toc" (a constant)"# divide terms on numerator/denominator by x #((5x)/x-15/x)/((2x)/x+4/x)=(5-15/x)/(2+4/x)# as #xto+-oo,f(x)to(5-0)/(2+0)# #rArry=5/2" is the asymptote"# Oblique asymptotes occur when the degree of the numerator > degree of the denominator.This is not the case here ( both of degree 1 ) Hence there are no oblique asymptotes. graph{(5x-15)/(2x+4) [-40, 40, -20, 20]}

Lily Anderson · Answer

undefined To find the vertical asymptote, set the denominator equal to zero and solve for x. In this case, the denominator is 2x + 4. So, 2x + 4 = 0 ⇒ x = -2. Therefore, the vertical asymptote is x = -2.

To find the horizontal asymptote, 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 y = 0. If the degree of the numerator is equal to the degree of the denominator, divide the leading coefficients of both polynomials to find the horizontal asymptote. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. In this case, the degree of the numerator is 1, and the degree of the denominator is also 1. So, the horizontal asymptote is the ratio of the leading coefficients, which is 5/2.

To find the oblique asymptote (also known as slant asymptote), perform long division or polynomial division to divide the numerator by the denominator. The quotient will represent the equation of the oblique asymptote. In this case, perform the division (5x - 15) ÷ (2x + 4). The result of the division will be the equation of the oblique asymptote.