Chapter 12, Problem 45RE

To determine

To find the limit (if it exists) by evaluating the corresponding one-sided limits..

Answer to Problem 45RE

Limit does not exist.

Explanation of Solution

Given information:

Graph the function. Determine the limit (if it exists) by evaluating the corresponding one-sided limits.

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

Calculation:

The limit of given function is,

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

Let plot the given function.

To calculate the model graphically use graphic utility TI-83calculator,

Press MATH select NUM and press Y key and enter the equation,

  

Press window button,

  

Press graph button,

  

From the graph we observe that the function converges to a single bounded valve as it approaches $3$ from both sides left and right.

  $\underset{x\to 3}{\lim }\frac{\left|x-3\right|}{x-3}=-1$

  $\underset{x\to 3}{\lim }\frac{\left|x-3\right|}{x-3}=+1$

Hence limit does not exist.