How do you find the vertical, horizontal and slant asymptotes of: #f(x)= ( x-3) /( x^2-3x+2)#?

Question

Cora Alexander · Accepted Answer

#H.A => y = 0#

#V.A =>x=1# and #x=2# Remember: You cannot have three asymptotes at the same time. If the Horizontal Asymptote exists, the Oblique/Slant Asymptote doesn't exist. Also, #color (red) (H.A)# #color (red) (follow)# #color (red) (three)# #color (red) (procedures).# Let's say #color (red)n# = highest degree of the numerator and #color (blue)m# = highest degree of the denominator,#color (violet) (if)#: #color (red)n color (green)< color (blue) m#, #color (red) (H.A => y = 0)# #color (red)n color (green)= color (blue) m#, #color (red) (H.A => y = a/b)# #color (red)n color (green)> color (blue) m#, #color (red) (H.A) # #color (red) (doesn't)# #color (red) (EE)# For this problem, #f(x)=(x-3)/(x^2-3x+2)# #color (red)n color (green)< color (blue) m#, #H.A => y = 0# #V.A=>x^2-3x+2=0# Find the answer by using the tools that you already know. As for me, I always use #Delta=b^2-4ac#, with #a=1#, #b=-3# and #c=2# #Delta=(-3)^2-4(1)(2)=1=>sqrt Delta=+-1# #x_1=(-b+sqrt Delta)/(2a)# and #x_2=(-b-sqrt Delta)/(2a)# #x_1=(3+1)/(2)=2# and #x_2=(3-1)/(2)=1# So, the #V.A# are #x=1# and #x=2# Hope this helps :)

Sadie Alexander · Answer

undefined To find the vertical asymptotes, set the denominator equal to zero and solve for x. In this case, the denominator $x^2 - 3x + 2$ factors to $(x - 1)(x - 2)$. So, the vertical asymptotes occur at $x = 1$ and $x = 2$.

To find the horizontal asymptote, compare the degrees of the numerator and the denominator. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $y = 0$.

To find the slant asymptote, perform polynomial long division or synthetic division of the numerator by the denominator. The quotient will give you the equation of the slant asymptote, if it exists. In this case, after performing the division, there is no slant asymptote because the degree of the numerator is not greater than the degree of the denominator by exactly 1.