Chapter 3.7, Problem 34E

To determine

the slant asymptote for the given function if exist, or give explanation if it does not exists.

Answer to Problem 34E

Slant asymptote does not exist because the degree of numerator is not one greater than that of denominator.

Explanation of Solution

Given information:

  $f\left(x\right)=\frac{x^3-4x^2+2x-6}{x+3}$

Calculation:

Divide the numerator of $f\left(x\right)$ by its denominator to get the quotient as slant asymptote.

  $x+3\begin{array}{c}\hfill x^2-7x+19\\ \hfill \overline{)\begin{array}{l}{x}^3-4x^2+2x-6\\ \underset{\_}{-\left(x^3-3x^2\right)}\downarrow \\ \underset{\_}{\begin{array}{l}-7x^2+2x\\ 7x^2-21x\downarrow \end{array}}\\ -19x-6\\ \underset{\_}{19x+57}\\ \text{ 51}\end{array}}\end{array}$

Thus, the parabolic asymptote is $y=x^2-7x+19$ which is quadratic in nature.

Since the degree of numerator is 2 greater than that of denominator. So, the function does not have a slant asymptote. Thus, the slant asymptote of given function does not exist.