# How do you find the vertical, horizontal or slant asymptotes for #(x^3-x+1)/(2x^4+x^3-x^2-1)#?

##### 1 Answer

This function has horizontal asymptote

It has no slant asymptotes or holes.

#### Explanation:

Given:

#f(x) = (x^3-x+1)/(2x^4+x^3-x^2-1)#

Note that the numerator has degree

We can deduce that this rational function has horizontal asymptote

By Descartes' Rule of Signs, we can tell that the denominator has one positive real zero, since the pattern of signs of its coefficients is

The pattern of signs of the coefficients of

These two real zeros are potentially vertical asymptotes. They can only fail to be so if the numerator is zero at them (which could result in a hole instead). This happens if the numerator and denominator have a common factor - so let's find their GCF using Euclid's method...

#2x^4+x^3-x^2-1 = (x^3-x+1)(2x+1)+x^2-x-2#

#x^3-x+1 = (x^2-x-2)(x+1)+2x+3#

#x^2-x-2 = (2x+3)(1/2x-5/4)+7/4#

Since we found no exact division, the GCF is constant. There is no common linear or higher degree factor.

So

Slant (a.k.a. oblique) asymptotes of a rational function can only arise if the degree of the numerator is one more than that of the denominator. Such is not the case in our example.

To actually find the zeros of

#x_1 ~~ -1.2029#

#x_2 ~~ 0.86170#

The graph of

graph{(x^3-x-1)/(2x^4+x^3-x^2-1) [-10, 10, -5, 5]}