How do you find the asymptotes for #(x-3)/(x-2)#?

Question

Elias Ackerman · Accepted Answer

Vertical asymptotes occur when the denominator of the 
rational function is #0#.  In this question this would occur when #x - 2 = 0#  i.e, #x = 2# [ Horizontal asymptotes can be found when the degree of the numerator and the degree of the denominator are equal. ] Here they are both of degree #1#  and so are equal. The horizontal asymptote is found by taking the ratio of leading coefficients . hence y# =1/1  = 1  #

Sadie Alexander · Answer

undefined To find the asymptotes for the rational function $ \frac{x - 3}{x - 2} $, we look at the behavior of the function as $ x $ approaches certain values.

1. Vertical Asymptote:
   Vertical asymptotes occur where the denominator of the function becomes zero, but the numerator does not. In this case, the vertical asymptote occurs when $ x - 2 = 0 $, which gives $ x = 2 $.

2. Horizontal Asymptote:
   To find the horizontal asymptote, we examine 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 the x-axis (y = 0). If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

In $ \frac{x - 3}{x - 2} $, both the numerator and denominator have a degree of 1. Therefore, the horizontal asymptote is the ratio of the leading coefficients, which are both 1. So, the horizontal asymptote is $ y = 1 $.

Therefore, the vertical asymptote is $ x = 2 $, and the horizontal asymptote is $ y = 1 $.