Chapter 11, Problem 33RE

(a)

To determine

To graph:

The limit value of the following functions:

  $\underset{x\to 3}{\lim }\frac{x-3}{x^2-9}$

Explanation of Solution

Given information:

  $\underset{x\to 3}{\lim }\frac{x-3}{x^2-9}$

Graph:

  

Interpretation:

The given function is:

  $\underset{x\to 3}{\lim }\frac{x-3}{x^2-9}$

The graph of the limit value at $x=3$ is drawn as above.

From the graph of the limit function $\underset{x\to 3}{\lim }\frac{x-3}{x^2-9}$ converges to single bounded value as it approaches $3$ both from left and right side.

(b)

To determine

To calculate:

The limit value of the following functions numerically:

  $\underset{x\to 3}{\lim }\frac{x-3}{x^2-9}$

Answer to Problem 33RE

The limit value of $\underset{x\to 3}{\lim }\frac{x-3}{x^2-9}$ is $\frac{1}{6}$ .

Explanation of Solution

Given information:

  $\underset{x\to 3}{\lim }\frac{x-3}{x^2-9}$

Calculation:

The given function is:

  $\underset{x\to 3}{\lim }\frac{x-3}{x^2-9}$

The limit of the function as $x\to 3$ is calculated as:

  $\begin{array}{l}\underset{x\to 3}{\lim }\frac{x-3}{x^2-9}=\underset{x\to 3}{\lim }\frac{\left(x-3\right)}{x^2-3^2}\\ \underset{x\to 3}{\lim }\frac{x-3}{x^2-9}=\underset{x\to 3}{\lim }\frac{\left(x-3\right)}{\left(x-3\right)\left(x+3\right)}\\ \underset{x\to 3}{\lim }\frac{x-3}{x^2-9}=\underset{x\to 3}{\lim }\frac{1}{\left(x+3\right)}\\ \underset{x\to 3}{\lim }\frac{x-3}{x^2-9}=\frac{1}{\left(3+3\right)}\left[\because x=3\right]\\ \underset{x\to 3}{\lim }\frac{x-3}{x^2-9}=\frac{1}{6}\\ \underset{x\to 3}{\lim }\frac{x-3}{x^2-9}=0.167\end{array}$

Hence, the limiting value is $\frac{1}{6}$ .