Chapter 2.3, Problem 25E

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 rises to the right.

Given:$f\left(x\right)=3x^4-5x^2+3$

Explanation for the end behaviour of $f\left(x\right)=3x^4-5x^2+3$:

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 $f\left(x\right)=3x^4-5x^2+3$ is a polynomial function of degree $4$ and the leading coefficients is $a_n>0$ .

  

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

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