How do you find the vertical, horizontal or slant asymptotes for #(x-2)/(x^2-4)#?

Question

Elijah Allen · Accepted Answer

vertical asymptote x = -2
horizontal asymptote y = 0

The first step we have to take here is to factorise and simplify the function. #rArr (x-2)/(x^2-4) = (x-2)/((x-2)(x+2) ) =cancel((x-2))/(cancel((x-2)) (x+2) # # = 1/(x+2)# Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation set the denominator equal to zero. solve : x + 2 = 0 → x = -2 is the asymptote Horizontal asymptotes occur as #lim_(x to+-oo) , f(x) to 0 # divide terms on numerator/denominator by x # rArr (1/x)/(x/x + 2/x) = (1/x)/(1 + 2/x)# as # x to +-oo , y to 0/(1+0)# #rArr y = 0 " is the asymptote " # Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here hence there are no slant asymptotes. graph{(x-2)/(x^2-4) [-10, 10, -5, 5]}

Sadie Ackerman · Answer

undefined To find the vertical, horizontal, or slant asymptotes for the function $ \frac{x-2}{x^2-4} $, follow these steps:

1. **Vertical Asymptotes**: Vertical asymptotes occur where the denominator of the fraction becomes zero, but the numerator doesn't. Set the denominator equal to zero and solve for $ x $. In this case, the denominator $ x^2 - 4 $ becomes zero when $ x = 2 $ or $ x = -2 $. Therefore, the vertical asymptotes are $ x = 2 $ and $ x = -2 $.

2. **Horizontal Asymptotes**: Horizontal asymptotes occur when $ x $ approaches positive or negative infinity. To find 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 $ y = 0 $. If the degrees are equal, divide the leading coefficients. If the degree of the numerator is greater, there is no horizontal asymptote. In this case, the degree of the numerator is 1 and the degree of the denominator is 2. Therefore, the horizontal asymptote is $ y = 0 $.

3. **Slant Asymptotes (Oblique Asymptotes)**: Slant asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator. To find slant asymptotes, perform polynomial long division of the numerator by the denominator. If there's a remainder, it will be a linear expression. The quotient will be the equation of the slant asymptote. In this case, $ x - 2 $ divided by $ x^2 - 4 $ results in $ \frac{1}{x+2} $ with a remainder of $ \frac{4}{x^2-4} $. The quotient $ \frac{1}{x+2} $ represents the slant asymptote.

So, the vertical asymptotes are $ x = 2 $ and $ x = -2 $, the horizontal asymptote is $ y = 0 $, and there is a slant asymptote given by $ y = \frac{1}{x+2} $.