Chapter 2.3, Problem 15E

Solution

To Sketch: The function pairs in the same series of viewing windows and state the final window view through zoom out.

Given: $f\left(x\right)=2x^3+3x^2-6x-15$ and $g\left(x\right)=2x^3$

Graph:

Draw the three views of the graphs $f\left(x\right)=2x^3+3x^2-6x-15$ and $g\left(x\right)=2x^3$ in increasingly larger viewing windows.

  

  

  

Interpretation:

When the dimensions of the window get larger, the graphs of $f\left(x\right)=2x^3+3x^2-6x-15$ and $g\left(x\right)=2x^3$ looks nearly each other.

Moreover,

  $\begin{array}{l}\underset{x\to \infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\lim }g\left(x\right)\\ \underset{x\to \infty }{\lim }f\left(x\right)=\infty \end{array}$

And

  $\begin{array}{l}\underset{x\to -\infty }{\lim }f\left(x\right)=\underset{x\to -\infty }{\lim }g\left(x\right)\\ \underset{x\to -\infty }{\lim }f\left(x\right)=-\infty \end{array}$

Thus, the limit values for the given function is $-\infty$ to $\infty$ and the functions of $f\left(x\right)=2x^3+3x^2-6x-15$ and $g\left(x\right)=2x^3$ nearly identical with each other when the window getting larger.