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

Question

Chloe Alexander · Accepted Answer

first factor out to simplest form:
#(x(x^2-1))/(x(x^2-4))=(x^2-1)/(x^2-4)#
You can further simplify it is of the form:
#(x^2-a^2) = (x+a)(x-a)#
And from here you should be able to find the asymptote
Good luck,
yonas

Lincoln Anderson · Answer

undefined To find the asymptotes of the function $ \frac{x^3 - x}{x^3 - 4x} $, follow these steps:

1. Determine the degree of the numerator and the denominator. Here, both the numerator and denominator have the same degree, which is 3.

2. If the degrees of the numerator and denominator are the same, divide the polynomial in the numerator by the polynomial in the denominator using polynomial long division or synthetic division.

3. After dividing, if the resulting quotient has no remainder, the asymptotes are the equations of the horizontal lines $ y = \frac{a}{c} $, where $ a $ is the leading coefficient of the numerator and $ c $ is the leading coefficient of the denominator.

4. If the resulting quotient has a remainder, there may be oblique (slant) asymptotes in addition to horizontal asymptotes. To find oblique asymptotes, perform polynomial long division or synthetic division again, but this time divide the numerator by the denominator and ignore the remainder. The quotient represents the equation of the oblique asymptote.

5. In this case, when you perform the division, you get a quotient of $ 1 $, so there are no oblique asymptotes. Therefore, the equation of the horizontal asymptote is $ y = \frac{1}{1} = 1 $.