Chapter 13.1, Problem 12E

To determine

As the limit of given function tends to $1$ therefore we will plot the table of values for $x\to 1^{-}$ and for $x\to 1^{+}$ to get the value of the limit.

Answer to Problem 12E

The limit as x approaches $1$ of $f(x)$ is $1\cdot 5$ as we can see from the table of values that as we approach toward $1$ the value of $f(x)$ will be gets towards $1\cdot 5$ . Also we can refer to graph for the limit, it will still be $1\cdot 5$ . So, our answer is verified graphically as well.

Explanation of Solution

Given information:

  $\underset{x\to 1}{\lim }\frac{x^3-1}{x^2-1}$

Calculation:

If we put $x=1$ we get $\frac{0}{0}$ form

x	$x\to 1^{-}$   $f(x)$
$0\cdot 9$	$1\cdot 426$
$0\cdot 99$	$1\cdot 492$
$0\cdot 999$	$1\cdot 506$

x	$x\to 1^{+}$   $f(x)$
$1\cdot 1$	$1\cdot 576$
$1\cdot 01$	$1\cdot 524$
$1\cdot 001$	$1\cdot 5074$

  $\therefore \underset{x\to 1}{\lim }\frac{x^3-1}{x^2-1}=1\cdot 5$

Graph:

Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expression $\underset{x\to 1}{\lim }\frac{x^3-1}{x^2-1}$ which is required to graph.

Step 4: Press GRAPH button to graph the function and adjust the windows according to the graph.

The graph is obtained as:

  

Interpretation:

The limit as x approaches $1$ of $f(x)$ is $1\cdot 5$ as we can see from the table of values that as we approach toward $1$ the value of $f(x)$ will be gets towards $1\cdot 5$ . Also we can refer to graph for the limit, it will still be $1\cdot 5$ . So, our answer is verified graphically as well.