Chapter 1, Problem 19RE

To determine

To find: Whether the $\underset{x\to b}{\lim }f\left(x\right)$ exists or not, for which the function $f\left(x\right)$ is shown below.

  

Answer to Problem 19RE

The value of $\underset{x\to b}{\lim }f\left(x\right)$ exists and it is finite.

Explanation of Solution

Given:

The graph for the function $\underset{x\to b}{\lim }f\left(x\right)$ is shown below.

  

Concept used:

The limit of the function $g\left(x\right)$ exists at some point $x=a$ in domain of $g\left(x\right)$ , if the function has no holes, jumps or breaks at that point $x=a$ in graph of function.

Calculation:

From the function $f\left(x\right)$ , it can be interpreted that the domain of the function $f\left(x\right)$ is real. And, it can be interpreted from the graph that as $x\to b^{-}$ (from left), the value of $\underset{x\to b^{-}}{\lim }f\left(x\right)$ is not defined because there exists a hole as $x\to b^{-}$ (from left). Similarly, when $x\to b^{+}$ (from right) the value of $\underset{x\to b^{+}}{\lim }f\left(x\right)$ is again undefined, because there exists a hole as $x\to b^{-}$ (from right).

Since the value of limit from left as $x\to b^{-}$ and limit from right as $x\to b^{+}$ are not defined. Therefore, $\underset{x\to b}{\lim }f\left(x\right)$ doesn’t exist.

Conclusion:

Hence, the value of $\underset{x\to b}{\lim }f\left(x\right)$ doesn’t exist.