Question

What are the asymptotes for `(2x+4)/x^3`?

Answer 1

V.A. @ `x=2`
H.A. @ `x=0`
No S.A.

Explanation:

The rules for horizontal asymptotes:

• When the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
• When the degree of the numerator is the same as the degree of the denominator, there is a horizontal asymptote at `x=0`
• When the degree of the numerator is less that the degree of the
  denominator, there is a horizontal asymptote at the quotient of the leading coefficients.

Because the denominator is larger, there is a horizontal asymptote at `x=2`

The rule for vertical asymptotes:

• There is a vertical asymptote at any value that will cause the function to be undefined.

Because `0` will cause the equation to be undefined (the denominator is equal to `0`), there is a vertical asymptote at `x=0`

The rule for slant asymptotes:

• When the degree of the numerator is exactly `1` more than the degree of the denominator, there is a slant asymptote at the quotient of the numerator and the denominator. (You have to divide the top by the bottom)

Because the denominator is larger, there is no slant asymptote.