Chapter 2, Problem 42RE

To determine

To graph: The function that satisfies the given conditions.

Explanation of Solution

Given information: The given conditions are $\underset{x\to 2}{\lim }f\left(x\right)$ does not exist and $\underset{x\to 2^{+}}{\lim }f\left(x\right)=f\left(2\right)=3$ .

Graph:

Consider that $\underset{x\to 2}{\lim }f\left(x\right)$ does not exist and $\underset{x\to 2^{+}}{\lim }f(x)=f\left(2\right)=3$ . So, the left hand limit and the right hand limit have different values.

It shows that the value of the function at $x=2$ is $3$ . Also, as $x\to 2$ approaches from right side then it is defined at $x=2$ . Mark a close point at $\left(2,3\right)$ .

As it is given that $\underset{x\to 2}{\lim }f\left(x\right)$ does not exist. So, the from the left side the limit is not defined at $x=2$ . Mark a hole at $x=2$ as $x\to 2$ approaches from the left side.

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

  

Figure (1)

Interpretation: From Figure (1), it can be observed that it a graph of a piecewise function that is not defined at $x=2$ .