How do you find vertical, horizontal and oblique asymptotes for #(5x-15)/(2x+4)#?

Question

Lily Anderson · Accepted Answer

Vertical asymptote: #x=-2#
Horizontal asymptote: #y=5/2#
There is not oblique asymptote.

Given #y=(5x-15)/(2x+4)# (after converting to an equation) As #(2x+4)rarr 0# #color(white)("XXX")yrarr +-oo# (whether it is plus or minus depends upon from which side #(2x+4)# approaches zero. #color(white)("XXX")2x+4=0# implies #x=-2# the vertical asymptote. The horizontal asymptote is given as the limit value of #y# as #xrarroo# #color(white)("XXX")lim_(xrarroo)y# #color(white)("XXX")=lim(xrarroo) (5x-15)/(2x+4)# #color(white)("XXX")=lim(xrarroo) (5-15/x)/(2+4/x)# #color(white)("XXX")=5/2# An oblique asymptote only exists if the degree of the numerator is greater than the degree of the denominator. #(5x-15)# and #(2x+4)# are both of degree #1#; so there is no oblique asymptote. graph{[2x+4)/(5x-15) [-16.86, 15.2, -7.67, 8.37]}

Elias Ackerman · Answer

undefined To find the vertical, horizontal, and oblique asymptotes of the function $ f(x) = \frac{5x - 15}{2x + 4} $, we can analyze the behavior of the function as $ x $ approaches infinity or negative infinity, as well as any vertical asymptotes due to division by zero.

**Vertical Asymptotes:** Vertical asymptotes occur where the denominator of the function is equal to zero, but the numerator is not. In this case, the denominator $ 2x + 4 $ is zero when $ x = -2 $, so there is a vertical asymptote at $ x = -2 $.

**Horizontal Asymptote:** To find the horizontal asymptote, we look at 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, the horizontal asymptote is the ratio of the leading coefficients. Here, the degree of the numerator is 1 and the degree of the denominator is 1, so the horizontal asymptote is the ratio of their leading coefficients: $ \frac{5}{2} = 2.5 $.

**Oblique Asymptote:** Oblique (or slant) asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. To find the oblique asymptote, perform polynomial long division or use synthetic division to divide the numerator by the denominator. The quotient will be the equation of the oblique asymptote. In this case, when we divide $ 5x - 15 $ by $ 2x + 4 $, we get $ 2.5x - 6.25 $, so the oblique asymptote is $ y = 2.5x - 6.25 $.

In summary:

- Vertical asymptote: $ x = -2 $
- Horizontal asymptote: $ y = 2.5 $
- Oblique asymptote: $ y = 2.5x - 6.25 $