How do you find all the asymptotes for function #(2x+4)/(x^2-3x-4)#?

Question

Benjamin Achtenberg · Accepted Answer

Factorise the denominator and examine the degrees of the numerator and denominator to find vertical asymptotes #x = -1#, #x = 4# and horizontal asymptote #y = 0#.

#f(x) = (2x+4)/(x^2-3x-4) = (2(x+2))/((x-4)(x+1))# This will have vertical asymptotes #x = -1# and #x = 4#. For both of these values of #x# the denominator is #0# and the numerator is non-zero. Since the degree #2# of the denominator is greater than the degree #1# of the numerator, we find: #f(x) -> 0# as #x -> +-oo# So there is a horizontal asymptote #y = 0#. graph{(2x+4)/(x^2-3x-4) [-10, 10, -5, 5]}

Elias Ackerman · Answer

undefined To find all the asymptotes for the function $ \frac{2x + 4}{x^2 - 3x - 4} $, we need to examine the behavior of the function as $ x $ approaches certain values.

1. **Vertical Asymptotes**:
   Vertical asymptotes occur where the denominator of the rational function becomes zero, but the numerator does not. To find them, solve the equation $ x^2 - 3x - 4 = 0 $ for $ x $.

2. **Horizontal Asymptotes**:
   Horizontal asymptotes occur when $ x $ approaches positive or negative infinity. To find horizontal asymptotes:
   - 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 to find the horizontal asymptote.
   - If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

3. **Oblique (Slant) Asymptotes**:
   Oblique asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator. To find oblique asymptotes, perform polynomial long division or synthetic division.

Once you find the vertical, horizontal, and oblique asymptotes, you will have identified all the asymptotes of the function.