Chapter 11.1, Problem 36E

To determine

To explain:

The limit of $\underset{x\to 3}{\lim }\frac{x-3}{\left|x+3\right|}$ if it exist and in case not exist; give reason.

Answer to Problem 36E

The limit of this function $\underset{x\to 3}{\lim }\frac{x-3}{\left|x+3\right|}$ doesn’t exist at $x=3$ .

Explanation of Solution

Given information:

  $\underset{x\to 3}{\lim }\frac{x-3}{\left|x+3\right|}$

  

Firstly to 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 $2$ different values as $x$ approaches from right and left.

Therefore, the limit of the function $\underset{x\to 3}{\lim }\frac{x-3}{\left|x+3\right|}$ doesn’t exist at $x=3$ .