Chapter 2, Problem 41RE

To determine

To graph: The function that satisfies the given conditions.

Explanation of Solution

Given information: The given function conditions are $\underset{x\to \infty }{\lim }f\left(x\right)=3$ , $\underset{x\to -\infty }{\lim }f\left(x\right)=\infty$ , $\underset{x\to 3^{+}}{\lim }f\left(x\right)=\infty$ , and $\underset{x\to 3^{-}}{\lim }\left(x\right)=-\infty$ .

Graph:

Consider $\underset{x\to \infty }{\lim }f\left(x\right)=3,$ it shows that $y=3$ acts as horizontals asymptote for the function on the positive side of $x$ . As x approaches to infinity, the value of the function is 3.

For $\underset{x\to -\infty }{\lim }f\left(x\right)=\infty$ , as x approaches to negative infinity then the function approaches to positive direction or $f\left(x\right)\to \infty$ .

For $\underset{x\to -3^{+}}{\lim }f\left(x\right)=\infty$ , as x approaches to 3 from the right side then the function approaches to $f\left(x\right)\to \infty$ .

For $\underset{x\to 3^{-}}{\lim }\left(x\right)=-\infty$ , as x approaches to 3 from the left side then the function approaches to $f\left(x\right)\to -\infty$ .

The graph of shown below shows the graph of a function that satisfies the given conditions.

  

Figure (1)

Interpretation: From the graph it can be observed that it a graph of a piecewise function including parabola, cubic function and exponential function.