Problem 36 - Precalculus: Mathematics for Calculus

Question

Find the limit, and use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 1} \frac{x^{8}-1}{x^{5}-1}$$

Step-by-Step Solution

Answer

The limit is \( \frac{7}{5} \).

Step 1: Identify the Limit Form

The expression given is \( \lim_{x \to 1} \frac{x^8-1}{x^5-1} \). Plugging in \( x = 1 \) yields \( \frac{0}{0} \), which is an indeterminate form. This suggests we need to simplify the expression to resolve the limit.

Step 2: Factor the Numerator and Denominator

We can factor both the numerator and denominator using the formula for the difference of powers. The numerator can be factored as \( x^8 - 1 = (x-1)(x^7+x^6+x^5+x^4+x^3+x^2+1) \). The denominator can be factored as \( x^5 - 1 = (x-1)(x^4+x^3+x^2+x+1) \).

Step 3: Cancel the Common Factor

Since both the numerator and denominator have a common factor of \( (x-1) \), we can cancel it. The simplified expression is \( \frac{x^7+x^6+x^5+x^4+x^3+x^2+1}{x^4+x^3+x^2+x+1} \).

Step 4: Substitute the Limit Value

Now, substituting \( x = 1 \) into the simplified expression gives us: \( \frac{1^7+1^6+1^5+1^4+1^3+1^2+1}{1^4+1^3+1^2+1+1} = \frac{7}{5} \).

Step 5: Graphical Verification

Graph \( \frac{x^8-1}{x^5-1} \) using a graphing device, such as a graphing calculator or software, and examine its value as \( x \) approaches 1. The graph should show that as \( x \) gets very close to 1, the function approaches 1.4, confirming our result.

Key Concepts

Indeterminate Form

When you're dealing with limits, sometimes plugging in the value straight away doesn't work, yielding what's called an "indeterminate form." This commonly happens with expressions resulting in forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). These forms are "indeterminate" because they don't provide clear information about the actual limit value. In the exercise given, substituting \( x = 1 \) into the function yields \( \frac{0}{0} \). While this does not determine our limit immediately, it signals the need to manipulate the function algebraically to find the limit's true value. Understanding indeterminate forms is important because they highlight situations where basic algebraic simplification is required to solve a problem correctly.

Factoring

Factoring is a technique used to simplify algebraic expressions, making it easier to solve equations or compute limits. In the given exercise, both the numerator \( x^8 - 1 \) and the denominator \( x^5 - 1 \) can be factored using the difference of powers formula:

• For \( x^8 - 1 \), the factorization is \( (x-1)(x^7+x^6+x^5+x^4+x^3+x^2+1) \).
• For \( x^5 - 1 \), it becomes \( (x-1)(x^4+x^3+x^2+x+1) \).

By applying these factorizations, we can understand how the original expression behaves near the limit point \( x = 1 \). Factorization is crucial in resolving indeterminate forms by revealing and canceling common factors in the numerator and denominator. This simplification is necessary to calculate the true limit.

Difference of Powers

The difference of powers refers to expressions like \( a^n - b^n \), which can specifically be factored into simpler components. In algebra, this method becomes especially helpful in simplifying complex expressions:

• The expression \( a^n - b^n \) can generally be factored as \( (a-b)(a^{n-1} + a^{n-2}b + \ldots + b^{n-1}) \) when \( n \) is a positive integer.

For instance, in the exercise, the polynomial expressions were excellent candidates for this method. The difference of powers provides us with a way to break down the expressions into product form, which is easier to manage analytically. For example:

• Factorizing \( x^8 - 1 \) and \( x^5 - 1 \) makes it possible to cancel out the common factor \( x-1 \), thus removing the indeterminate form and allowing the limit to be evaluated more easily.

Graphical Verification

Graphical verification involves using a graphing tool to visually confirm the behavior of a function as it approaches a certain point or limit. In the exercise, after simplifying the problem algebraically, graphing provides an empirical validation of the result:

• Plotting the function \( \frac{x^8 - 1}{x^5 - 1} \) shows how it behaves near \( x = 1 \).

By observing the graph, students can verify that the function approaches the calculated limit value of 1.4 as \( x \) nears 1. It helps reassure that algebraic manipulation was done correctly. Employing a graphing device serves as a practical tool for visual learners and can help students intuitively grasp complex mathematical behavior.