Problem 36 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits. \(\lim _{x \rightarrow 2^{+}} \frac{x^{2}+2 x-8}{x^{2}-4}\)

Step-by-Step Solution

Answer

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

Step 1: Analyze the Expression

The expression given is \( \frac{x^2 + 2x - 8}{x^2 - 4} \). We need to find the limit as \( x \) approaches 2 from the positive side, denoted as \( x \rightarrow 2^+ \). First, observe that the denominator \( x^2 - 4 \) can be factored further to test for simplification.

Step 2: Factor and Simplify

The denominator \( x^2 - 4 \) is a difference of squares, which can be written as \( (x - 2)(x + 2) \). Likewise, factor the numerator \( x^2 + 2x - 8 \) to \( (x - 2)(x + 4) \). Thus, the expression simplifies to \( \frac{(x - 2)(x + 4)}{(x - 2)(x + 2)} \).

Step 3: Cancel Common Factors

Since both the numerator and the denominator have \( (x - 2) \) as a factor, cancel this common factor. The simplified expression becomes \( \frac{x + 4}{x + 2} \).

Step 4: Evaluate the Limit on Simplified Expression

Substitute \( x = 2 \) into the simplified expression \( \frac{x + 4}{x + 2} \) to find the limit: \( \frac{2 + 4}{2 + 2} = \frac{6}{4} = \frac{3}{2} \).

Step 5: Conclusion of the Limit

Since there are no undefined behaviors (like division by zero) in the simplified expression near \( x = 2 \), the limit from the positive side is \( \frac{3}{2} \).

Key Concepts

Factorization

Factorization is a technique often used to simplify complex algebraic expressions. By factorizing, you break down an expression into simpler terms or factors that, when multiplied together, produce the original expression. This process is very useful in mathematics when you need to simplify and solve algebraic equations, especially in calculus when evaluating limits.

• The first step is to look for common factors that can be easily identified and factored out.
• In the case of the given exercise, the expression \(x^2 - 4\) is a difference of squares and can be factored into \( (x - 2)(x + 2) \).
• Similarly, the expression \(x^2 + 2x - 8\) can be factorized into \( (x - 2)(x + 4) \) by applying techniques like trial and error or the quadratic formula.

Factorization allows for greater simplification, making it easier to analyze the behavior of a function as it approaches a specific value.

Simplification

Simplification is the process of reducing a mathematical expression to its most straightforward form. This is crucial when evaluating limits because it can help remove complexities that might obscure the limit's evaluation.

• After factorizing both the numerator and the denominator as seen in the original exercise, the original expression becomes \((x - 2)(x + 4)/(x - 2)(x + 2)\).
• By simplifying this, you focus only on the essential parts, leading to an expression free of factors that could make the expression problematic.
• This new form helps in smoothly evaluating the limit by removing any ambiguity or zero over zero forms.

Simplification streamlines the problem-solving process, making it easier to see and understand the core behavior of the function under investigation.

Difference of Squares

The difference of squares is a specific way of recognizing and factorizing expressions through their unique structure. It refers to a particular formula that boils down to the subtraction of two squared terms.

• The general form for a difference of squares is \(a^2 - b^2 = (a - b)(a + b)\).
• In our expression, \(x^2 - 4\) is a difference of squares because it can be rewritten as \(x^2 - 2^2\).
• Factoring using the difference of squares results in the components \( (x - 2)(x + 2) \), which simplifies further calculations.

Recognizing patterns like the difference of squares helps to quickly factorize expressions and simplify limits in calculus efficiently.

Canceling Common Factors

Canceling common factors is a crucial step when simplifying an algebraic expression. It involves removing the same terms from the numerator and denominator to simplify the expression further.

• In the given exercise, after factorizing, the expression \( (x - 2)(x + 4)/(x - 2)(x + 2) \) has \(x - 2\) as a common factor.
• Canceling \(x - 2\) from both the numerator and the denominator eliminates any potential undefined operations, such as dividing by zero.
• After cancelation, the expression simplifies to \(x + 4)/(x + 2)\), allowing for straightforward evaluation of the limit.

Canceling common factors helps in ensuring that the limit does not involve any indeterminate forms, thus enabling a clean calculation of the limit's value.