Chapter 11.1, Problem 49E

To determine

To graph:

The following function and its limit value and in case there is no limit; give reason.

  $f\left(x\right)=\ln \left(x+3\right),\underset{x\to 4}{\lim }f\left(x\right)$

Explanation of Solution

Given information:

  $f\left(x\right)=\ln \left(x+3\right),\underset{x\to 4}{\lim }f\left(x\right)$

Graph:

  

Interpretation:

Firstly determine limit at specified point. Then check the limit exists at the specified point by the points.

Some conditions for existence of the limits:

When the function satisfies the conditions, then limit of function doesn’t exist.

1. $f\left(x\right)$ increases or decreases without a bound as $x$ approaches to $c$ .
2. $f\left(x\right)$ approaches the different number from right side of the $c$ than its approach from left side of the $c$ .
3. $f\left(x\right)$ oscillates between $2$ fixed values as $x$ approaches to $c$ .

From this graph, the function value approaches two different values as $x$ approaches from right and left as:

  $\begin{array}{l}\underset{x\to +4}{\lim }\ln \left(x+3\right)=\ln \left(4+3\right)=\ln 7\\ \underset{x\to -4}{\lim }\ln \left(x+3\right)=\ln \left(-4+3\right)=\ln \left(-1\right)\end{array}$

So, on checking three conditions, the given function does not satisfy any of the conditions.

Now, the limit of the function $f\left(x\right)$ is calculated as:

  $\begin{array}{l}\underset{x\to 4}{\lim }f\left(x\right)=\underset{x\to 4}{\lim }\ln \left(x+3\right)\\ \underset{x\to 4}{\lim }\ln \left(x+3\right)=\ln \left(4+3\right)\\ \underset{x\to 4}{\lim }\ln \left(x+3\right)=\ln 7\end{array}$