How do you find vertical, horizontal and oblique asymptotes for # f(x) = (x-4)/ (x^2-1)#?

Question

Sadie Ackerman · Accepted Answer

vertical asymptotes x = ± 1
horizontal asymptote y = 0

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation, equate the denominator to zero. solve : # x^2 - 1 = 0 → (x - 1 )(x + 1 ) = 0 # #rArr x = ± 1 " are the asymptotes " # Horizontal asymptotes occur as #lim_(x→±∞) f(x) → 0 # If the degree of the numerator is less than the degree of the denominator , as in this case, degree of numerator 1 and degree of denominator 2 . Then the equation is always y = 0. Here is the graph of the function. graph{(x-4)/(x^2-1) [-10, 10, -5, 5]}

Madelyn Anderson · Answer

undefined To find the vertical, horizontal, and oblique asymptotes for $ f(x) = \frac{x - 4}{x^2 - 1} $, follow these steps:

Vertical asymptotes:
1. Determine the values of $ x $ that make the denominator zero.
2. These values will be the vertical asymptotes.

Horizontal asymptotes:
1. Compare the degrees of the numerator and the denominator.
2. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $ y = 0 $.
3. If the degree of the numerator is equal to the degree of the denominator, divide the leading coefficients of the numerator and denominator to find the horizontal asymptote.

Oblique asymptotes:
1. If the degree of the numerator is exactly one greater than the degree of the denominator, there is an oblique asymptote.
2. Use long division or polynomial division to divide the numerator by the denominator.
3. The quotient obtained will represent the equation of the oblique asymptote.