Chapter 6.3, Problem 65HP

To determine

To Find:The end behaviour of $\frac{f\left(x\right)}{g\left(x\right)}$ and explain the reasoning.

Answer to Problem 65HP

The end behaviour is $\frac{f\left(x\right)}{g\left(x\right)}\to -\infty \text{as}x\to +\infty$ and $\frac{f\left(x\right)}{g\left(x\right)}\to +\infty \text{as}x\to -\infty$ .

Explanation of Solution

Given:

The $f\left(x\right)$ has the degree 5 and the positive leading coefficient and $g\left(x\right)$ has the degree of 3 and the positive leading coefficient.

Calculation:

Consider the given function is,

  $\frac{f\left(x\right)}{g\left(x\right)}$

The degree of the above function is $5-3=2$ .

Consider that the positive degree is divided by the positive degree and will remain the positive degree.

Since, the degree is even and the leading coefficient is positive. Thus, the end behaviour is,

  $\begin{array}{l}\frac{f\left(x\right)}{g\left(x\right)}\to -\infty \text{as}x\to +\infty \\ \frac{f\left(x\right)}{g\left(x\right)}\to +\infty \text{as}x\to -\infty \end{array}$