Problem 8 - Precalculus: Mathematical for Calculus

Question

Use a table of values to estimate the value of the limit. Then use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 1} \frac{x^{3}-1}{x^{2}-1}$$

Step-by-Step Solution

Answer

The limit is \( \frac{3}{2} \).

Step 1: Understanding the Function

The given function is \( f(x) = \frac{x^3 - 1}{x^2 - 1} \). As \( x \) approaches 1, the expression appears to become \( \frac{0}{0} \), which is an indeterminate form. We will need to simplify it or use another method to evaluate the limit.

Step 2: Creating a Table of Values

To estimate the limit, we create a table with values of \( x \) approaching 1 from both sides (e.g., 0.9, 0.99, 0.999, 1.001, 1.01, 1.1). For each \( x \) value, compute \( f(x) \). Observe how \( f(x) \) changes as \( x \) gets closer to 1.

Step 3: Simplifying the Function

Factor the numerator and denominator: \( x^3 - 1 = (x-1)(x^2+x+1) \) and \( x^2 - 1 = (x-1)(x+1) \). Simplify: \[ \frac{(x-1)(x^2+x+1)}{(x-1)(x+1)} = \frac{x^2+x+1}{x+1} \] since \( x eq 1 \), and observe this is valid when \( x eq 1 \).

Step 4: Calculating the Limit

Use the simplified form \( \frac{x^2+x+1}{x+1} \). Substitute \( x = 1 \) to find the limit: \( \frac{1^2+1+1}{1+1} = \frac{3}{2} \).

Step 5: Using a Graphing Device

Use a graphing calculator or software to graph \( y = \frac{x^3 - 1}{x^2 - 1} \). Observe the behavior of the graph as \( x \) approaches 1. The value the function approaches should confirm the calculated limit of \( \frac{3}{2} \).

Key Concepts

Factoring Polynomials

When dealing with limits that present us with an indeterminate form like \( \frac{0}{0} \), factoring polynomials can be an essential technique to simplify the expression. In our given problem, the function is \( \frac{x^3 - 1}{x^2 - 1} \). Both the numerator and the denominator can be factored.
- **Numerator**: \( x^3 - 1 \) can be expressed as a difference of cubes, which factors into \( (x-1)(x^2+x+1) \).
- **Denominator**: \( x^2 - 1 \), a difference of squares, factors into \( (x-1)(x+1) \).
With this factorization, the function simplifies significantly once \( x eq 1 \):
\[ \frac{x^3 - 1}{x^2 - 1} = \frac{(x-1)(x^2+x+1)}{(x-1)(x+1)} = \frac{x^2+x+1}{x+1} \]
After canceling \( (x-1) \) from both the numerator and the denominator. This simplification step is legitimate for any \( x \) except \( 1 \), which helps in dealing with the indeterminate form and estimating the limit easily.

Indeterminate Forms

The expression \( \frac{x^3 - 1}{x^2 - 1} \) produces an indeterminate form \( \frac{0}{0} \) when \( x = 1 \). Indeterminate forms arise in calculus when substituting a point into a function results in expressions like \( \frac{0}{0} \), \( \infty - \infty \), or \( 0 \times \infty \). These forms require additional techniques to understand the behavior of the function around these points.
In this exercise, to resolve the indeterminate form at \( x = 1 \), we factor the numerator and denominator to simplify the expression, as shown in the previous section. Once simplified, the expression is no longer indeterminate for \( x eq 1 \).
Understanding and dealing with indeterminate forms is crucial in calculus since they often arise when calculating limits. The goal is to transform the expression into a determinate form that allows us to evaluate the limit directly, or to use numerical or graphical methods for approximation.

Graphing Functions

Graphing functions provides a visual representation of the behavior of a function as the variable approaches a certain point. For the function \( f(x) = \frac{x^3 - 1}{x^2 - 1} \), graphing helps confirm the limit we found analytically. By plotting this function, we observe how it behaves as \( x \) approaches 1.
Using a graphing calculator or graphing software:
- Plot the simplified function \( \frac{x^2+x+1}{x+1} \).
- Examine the graph near \( x = 1 \).
The graph will approach a particular y-value as \( x \) gets closer to 1 from both sides. This y-value should corroborate our calculated limit of \( \frac{3}{2} \). Visualizing the function in this way not only validates our algebraic work but also provides intuition about continuity and behavior of functions around points of interest.
Graphs are powerful tools in calculus for interpreting results and understanding why certain algebraic manipulations (like factoring) help resolve indeterminate forms into limits that are understandable and meaningful.