Problem 18 - Precalculus with Limits

Question

In Exercises 9-36, find the limit (if it exists). Use a graphing utility to verify your result graphically. $$\lim_{x \to 1} \dfrac{x^4-1}{x-1}$$

Step-by-Step Solution

Answer

The limit of the function as x approaches 1 is 4.

Step 1: Factorization

Start by factorising the numerator of the function, you will use difference of squares twice. The difference of squares \(a^2 - b^2\) factored is \( (a-b)(a+b)\). So \(x^4-1 = (x^2+1)(x^2-1) = (x^2+1)(x-1)(x+1)\). Replace this in the function, so \(f(x) = \dfrac{(x^2+1)(x-1)(x+1)}{x-1}\). Now we can cancel out \(x-1\) from the numerator and denominator, and reduce the equation to \(f(x)= (x^2+1)(x+1)\)

Step 2: Find the limit

Substitute x with 1 in the reduced function, to obtain the limit \(L = \lim_{x \to 1} (x^2+1)(x+1) = (1^2+1)(1+1) = 2 \cdot 2 = 4\)

Step 3: Graphical verification

Using a graphing utility, plot the function \( (x^2+1)(x+1)\). Observe that as x approaches 1, the y-value of the function approaches 4, which confirms the result obtained in the previous step.

Key Concepts

Difference of Squares

In mathematics, the difference of squares is a specific pattern of factoring. The formula for the difference of squares is: \\[ a^2 - b^2 = (a - b)(a + b) \]This pattern allows us to factor expressions where two squares are subtracted. In the given exercise, the term \( x^4 - 1 \) represents a difference of squares. Let's break it down:

• Observe that \( x^4 \) can be viewed as \((x^2)^2\) and \(1\) as \(1^2\).
• Applying the difference of squares formula, we get: \( (x^2)^2 - (1)^2 = (x^2-1)(x^2+1) \).

But we can go a step further. Notice that \( x^2 - 1 \) is again a difference of squares:

• Here, \( x^2 - 1 = (x-1)(x+1) \) by the same formula.

By employing the difference of squares twice, the expression \( x^4 - 1 \) factors entirely into \((x-1)(x+1)(x^2+1)\). This is a powerful technique that greatly simplifies the problem at hand.

Factoring Polynomials

Factoring polynomials is a fundamental skill in calculus and algebra that simplifies complex expressions. It involves breaking down a polynomial into a product of simpler polynomials. In this exercise, after applying the difference of squares, we achieved a factored form: \\[ x^4 - 1 = (x^2+1)(x-1)(x+1) \]The key step in solving the limit problem is simplifying the expression by canceling common factors in the numerator and denominator. Here's how it works:

• The original limit expression is \( \frac{x^4-1}{x-1} \).
• After factoring the numerator, the expression becomes \( \frac{(x^2+1)(x-1)(x+1)}{x-1} \).
• The \( x-1 \) in the numerator and denominator cancels out, leaving \( f(x) = (x^2+1)(x+1) \).

This reduced polynomial omits the problematic zero-division point, allowing the limit to be evaluated directly. Factoring polynomials reduces complexity and removes indeterminate forms, such as \( \frac{0}{0} \), that might otherwise cause difficulty in calculating limits.

Graphical Verification

Graphical verification is an excellent method for confirming algebraic results visually. It involves using a graphing tool to plot a function and observing its behavior as the variable approaches a particular value. In this problem:

• The reduced function to verify is \( (x^2+1)(x+1) \).
• Using a graphing utility, you can plot this function and analyze its graph as \( x \) approaches 1.
• The value the function approaches is 4, which matches the calculated limit.

Through graphical verification, you gain a visual understanding of the function's behavior, especially near points of interest like \( x = 1 \). This process reassures you that the algebraic methods used previously are correct: seeing the limit value on the graph acts as a double-check for potential computational errors made while evaluating the limit. Graphical analysis complements algebraic solutions by providing an intuitive and observable insight into mathematical concepts.