Question

What are the vertical and horizontal asymptotes for the function f (x) = StartFraction x squared + x minus 6 Over x cubed minus 1 EndFraction?

Answer 1

b on edge 2021

Explanation:

Answer 2

Given:

The given function is
`f(x)=(x^(2)+x-6)/(x^(3)-1)`

We need to determine the horizontal and vertical asymptote.

Horizontal asymptote:

From the given function, it is obvious that the denominator's degree is greater than the numerator's degree.

Then, the horizontal asymptote is the x - axis.

Thus, the horizontal asymptote is
`y=0`

Vertical asymptote:

The vertical asymptote of the rational function are the undefined points and can be determined by equating the denominator equal to zero.

Thus, we have;

`x^3-1=0`

`x^3=1`

Solving, we get;

`x=1, x=(-1+√(3) i)/(2), x=(-1-√(3) i)/(2)`

Thus, the function is undefined at the point
`x=1`

Hence, the vertical asymptote of the function is
`x=1`