Chapter 2.3, Problem 27E

Solution

To Explain: The end behaviour of the given polynomial using the limits $\underset{x\to \infty }{\lim }f\left(x\right)$ and $\underset{x\to -\infty }{\lim }f\left(x\right)$ .

The polynomial has the limit of $\underset{x\to -\infty }{\lim }f\left(x\right)=\infty$ and $\underset{x\to \infty }{\lim }f\left(x\right)=-\infty$ with the graph rises to the left and falls to the right.

Given:$f\left(x\right)=7x^2-x^3+3x-4$

Explanation for the end behaviour of $f\left(x\right)=7x^2-x^3+3x-4$:

For any polynomial function $f\left(x\right)=a_n{x}^n+\dots +a_{1}x+a_0$ , the limits $\underset{x\to \infty }{\lim }f\left(x\right)$ and $\underset{x\to -\infty }{\lim }f\left(x\right)$ can be found by the degree $n$ of the polynomial and its leading coefficients $a_n.$

Here, the function $f\left(x\right)=7x^2-x^3+3x-4$ is a polynomial function of degree $3$ and the leading coefficients is $a_n<0$ .

  

And the graph rises to the left and falls to the right.

Therefore, $\underset{x\to -\infty }{\lim }f\left(x\right)=\infty$ and $\underset{x\to \infty }{\lim }f\left(x\right)=-\infty$ .