How do you find the Vertical, Horizontal, and Oblique Asymptote given #y = (2x+4)/( x^2-3x-4)#?

Question

Theodore Amidon · Accepted Answer

so we have vertical asymptotes at # x = -1, 4#

for horizontal and slope asmptotes, #lim_{x o pm oo} y = 0#

for vertical aympptotes, we look at when the demoninator is zero

so
#x^2 - 3x - 4 = (x+ 1)(x - 4) implies x = -1, 4#

to check for possible indeterminates we note that

#y(-1) = 2/0# = ndef
and
#y(4) = 12/0# = ndef

so we have vertical asymptotes at # x = -1, 4#

for horizontal and slope we look at the behaviour of the function as #x o pm oo#

so we re-write
#lim_{x to pm oo} (2x+4)/(x^2 - 3x - 4)#

as

#lim_{x to pm oo} (2/x+4/x^2)/(1 - 3/x - 4/x^2) = 0#

Elijah Allen · Answer

undefined To find the vertical, horizontal, and oblique asymptotes of the function $y = \frac{2x + 4}{x^2 - 3x - 4}$, follow these steps:

1. **Vertical Asymptotes**:
   Vertical asymptotes occur where the denominator of the function is equal to zero, but the numerator is not zero.
   Set the denominator $x^2 - 3x - 4$ equal to zero and solve for $x$ to find the vertical asymptotes.

2. **Horizontal Asymptotes**:
   Horizontal asymptotes occur when $x$ approaches positive or negative infinity. To find horizontal asymptotes, 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 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 Asymptotes**:
   Oblique (slant) asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator.
   To find the oblique asymptote, perform polynomial long division or use another method to divide the numerator by the denominator. The quotient represents the equation of the oblique asymptote.

Following these steps will help you determine the vertical, horizontal, and oblique asymptotes for the given function.