Chapter 11.3, Problem 6QR

Solution

To find: The end behavior asymptote of $f\left(x\right)=\frac{x^4+2x^3+x+1}{x-3}$

The end behavior asymptote of $f\left(x\right)=\frac{x^4+2x^3+x+1}{x-3}$ is $y=x^3$ , option (c).

Given information:

The given function is $f\left(x\right)=\frac{x^4+2x^3+x+1}{x-3}$ .

Calculation:

Consider the function $f\left(x\right)=\frac{x^4+2x^3+x+1}{x-3}$

This function can be rewritten as

  $f\left(x\right)=x^3+5x^2+15x+6+\frac{139}{x-3}$

  $\underset{x\to \infty }{\lim }\frac{139}{x-3}=0$

Thus end behavior of $f\left(x\right)$ depends on $x^3+5x^2+15x+6$ .

  $\begin{array}{c}\underset{x\to \infty }{\lim }\left(x^3+5x^2+15x+6\right)=\underset{x\to \infty }{\lim }{x}^{{}^3}\left(1+\frac{5}{x}+\frac{15}{x^2}+\frac{6}{x^3}\right)\\ \underset{x\to \infty }{\lim }\left(x^3+5x^2+15x+6\right)=\underset{x\to \infty }{\lim }{x}^{{}^3}\end{array}$

Thus, the end behavior asymptote of $f\left(x\right)=\frac{2x^3-3x^2+1}{3-x}$ is $y=x^3$ .

Hence, option (c) is correct.