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

Given:

The given function is $f(x)=\frac{x^{2}+x-6}{x^{3}-1}$

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;

$x=1, x=\frac{-1+\sqrt{3} i}{2}, x=\frac{-1-\sqrt{3} i}{2}$

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

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