Question

Use a table of values to estimate the value of the limit. Then use a graphing device to confirm your result graphically.

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

Short answer

The value of limit of the given function is$\frac{3}{2}$.

Step-by-step solution

Step 1. Given information.

The limit of function here given is,

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

Step 2. Formula used.

The algebraic formula will be used here to simplify.

That is,$a^2-b^2=\left(a+b\right)\left(a-b\right)$

And,$a^3-b^3=\left(a-b\right)(a^2+ab+b^2)$

Step 3. Simplification.

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

Using the formula we get,

$\begin{array}{c}\underset{x\to 1}{\lim }\frac{{\left(x\right)}^3-{\left(1\right)}^3}{{\left(x\right)}^2-{\left(1\right)}^2}\\ \underset{x\to 1}{\lim }\frac{(x-1)(x^2+x+1)}{(x+1)(x-1)}\\ \underset{x\to 1}{\lim }\frac{(x^2+x+1)}{(x+1)}\end{array}$

Step 4. Make the table.

Now, make a table by putting the value of $x$.

Therefore, on the basis of the values in the table we make the guess that,

$\underset{x\to 1}{\lim }\frac{x^3-1}{x^2-1}=1.50=\frac{3}{2}$

Step 5. Draw the graph.

Now, as a graphical verification use a graphing device.

From the graph it is verified that,$\underset{x\to 1}{\lim }\frac{x^3-1}{x^2-1}=\frac{3}{2}$.