Chapter 11, Problem 1RE

To determine

The limit of the expression $\underset{x\to 1}{\lim }\frac{x^3-1}{x-1}$ by constructing a table.

Answer to Problem 1RE

Solution:

The value of the expression $\underset{x\to 1}{\lim }\frac{x^3-1}{x-1}$ is 3.

Explanation of Solution

Given Information:

Consider the provided function is $f\left(x\right)=\frac{x^3-1}{x-1}$

Consider the provided expression is $\underset{x\to 1}{\lim }\frac{x^3-1}{x-1}$.

As $x$ gets closer to 1 but remains unequal to 1,

Construct a table representing the number that the corresponding values of $\frac{x^3-1}{x-1}$ get closer to.

Choose values of $x$ closer to $1$ from the left side.

Take $x=0.99$

and find the corresponding value of $f\left(x\right)=\frac{x^3-1}{x-1}$.

Thus,

$\begin{array}{c}f\left(0.99\right)=\frac{{0.99}^3-1}{0.99-1}\\ =2.9701\end{array}$

Choose an additional value of $x$ that is closer to $1$ but still less than $1$.

Take $x=0.999$

and find the corresponding value of $f\left(x\right)=\frac{x^3-1}{x-1}$.

Thus,

$\begin{array}{c}f\left(0.999\right)=\frac{{0.9999}^3-1}{0.9999-1}\\ =2.99970001\end{array}$

Choose an additional value of $x$ that is closer to $1$ but still less than $1$.

Take $x=0.9999$

and find the corresponding value of $f\left(x\right)=\frac{x^3-1}{x-1}$.

Thus,

$\begin{array}{c}f\left(0.9999\right)=\frac{{0.9999}^3-1}{0.9999-1}\\ =2.99970001\end{array}$

List the values of $x$ closer to 1 as $x$ approaches from the left.

$x$	$f\left(x\right)=\frac{x^3-1}{x-1}$
0.99	2.9701
0.999	2.997001
0.9999	2.99970001

From the above table, it is clear that, as $x$ gets closer to 1 from the left side, the values of $f\left(x\right)$ get closer to 3.

Now choose values of $x$ closer to $1$ from the right side.

Take $x=1.01$

and find the corresponding value of $f\left(x\right)=\frac{x^3-1}{x-1}$.

Thus,

$\begin{array}{c}f\left(1.01\right)=\frac{{1.01}^3-1}{1.01-1}\\ =3.0301\end{array}$

Choose an additional number of $x$ that is closer to 1 but still greater than 1.

Take $x=1.001$

and find the corresponding value of $f\left(x\right)=\frac{x^3-1}{x-1}$.

Thus,

$\begin{array}{c}f\left(1.001\right)=\frac{{1.001}^3-1}{1.001-1}\\ =3.003001\end{array}$

Again, select an additional number of $x$ that is closer to 1 but still greater than 1.

Take $x=1.001$

and find the corresponding value of $f\left(x\right)=\frac{x^3-1}{x-1}$.

Thus,

$\begin{array}{c}f\left(1.0001\right)=\frac{{1.0001}^3-1}{1.0001-1}\\ =3.00030001\end{array}$

List the values of $x$ closer to 1 as $x$ approaches from the right.

$x$	$f\left(x\right)=\frac{x^3-1}{x-1}$
1.01	3.0301
1.001	3.003001
1.0001	3.00030001

From the above table, it is clear that, as $x$ gets closer to 1 from the right side, the values of $f\left(x\right)$ get closer to 3.

Combine both the tables as follows:

$x$ approaches 1 from the left	$\begin{array}{l}\text{Corresponding values}\\ \text{of }f\left(x\right)=\frac{x^3-1}{x-1}\\ \text{when }x\text{ approaches 1}\\ \text{from the left}\end{array}$	$\begin{array}{l}\text{Corresponding values}\\ \text{of }f\left(x\right)=\frac{x^3-1}{x-1}\\ \text{when }x\text{ approaches 1}\\ \text{from the right}\end{array}$	$x$ approaches 1 from the right
0.99	2.9701	3.0301	0.99
0.999	2.997001	3.003001	0.999
0.9999	2.99970001	3.00030001	0.9999

The tables show the values of $\frac{x^3-1}{x-1}$, as $x$ approaches 1 from left and right directions.

Thus as $x$ gets closer to 1 from both the directions, the values of $f\left(x\right)$

get closer to 3.

Therefore, $\underset{x\to 1}{\lim }\frac{x^3-1}{x-1}=3$