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 \( \frac{3}{2} \) as \( x \to 2^+ \).

Step 1: Analyze the Function

The given function is \( \frac{x^{2}+2x-8}{x^{2}-4} \). Notice that both the numerator and denominator are quadratic polynomials. The limit is as \( x \to 2^+ \), so we need to determine the behavior of the function close to \( x = 2 \) from the right-hand side.

Step 2: Simplify the Denominator

The denominator \( x^{2}-4 \) can be factored using the difference of squares formula: \( x^{2} - 4 = (x - 2)(x + 2) \). This helps us understand the form of the expression better.

Step 3: Simplify the Numerator

The numerator \( x^{2} + 2x - 8 \) can be factored. Factoring the quadratic equation, we find it is \( (x - 2)(x + 4) \). Substituting back, the expression becomes \( \frac{(x-2)(x+4)}{(x-2)(x+2)} \).

Step 4: Cancel Common Terms

Since both the numerator and the denominator have a common factor \( (x - 2) \), they can be canceled out (but only for \( x eq 2 \)). After canceling, we have the simplified function \( \frac{x+4}{x+2} \) for \( x eq 2 \).

Step 5: Find the Limit

Now, we evaluate the limit of the simplified function \( \frac{x+4}{x+2} \). Substitute \( x = 2 \) into the expression (noting that we're approaching from the right): \( \lim_{x \to 2^+} \frac{x+4}{x+2} = \frac{2+4}{2+2} = \frac{6}{4} = \frac{3}{2} \).

Key Concepts

Right-Hand Limit

When discussing limits, a right-hand limit is a concept used to describe the behavior of a function as the input approaches a certain point specifically from the right-hand side. In mathematical notation, this is expressed as \( \lim_{x \to c^+} f(x) \), which means that we are interested in the values that \( f(x) \) takes as \( x \) gets arbitrarily close to \( c \) from values that are slightly greater than \( c \).
Understanding right-hand limits is crucial when dealing with points where the function might become undefined or behave erratically due to division by zero or other factors. For the exercise in question, we consider the limit of the function \( \frac{x^2 + 2x - 8}{x^2 - 4} \) as \( x \) approaches 2 from values greater than 2.
This means we look at the behavior of the function for inputs like 2.1, 2.01, 2.001, closely approaching the point from the right.

Factoring Quadratics

Factoring quadratics is a method used to simplify quadratic expressions, making it easier to analyze functions, especially when dealing with limits. A quadratic expression is generally in the form \( ax^2 + bx + c \).

• Difference of Squares: One commonly used technique is recognizing a difference of squares. For instance, \( x^2 - 4 \) can be simplified to \((x - 2)(x + 2)\). This applies when the quadratic can be expressed as \( (x - a)(x + a) \).

In this exercise, both the numerator \( x^2 + 2x - 8 \) and the denominator \( x^2 - 4 \) are quadratics. The numerator can be factored into \((x - 2)(x + 4)\). By simplifying these expressions, we can better understand potential discontinuities and points to consider as \(x \to 2^+\).
Overall, factoring quadratics helps us break down complex functions into simpler, more manageable terms, revealing hidden behaviors as limits approach.

Simplifying Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Simplifying these expressions is crucial, especially when analyzing limits close to points where they may not initially appear defined.
To simplify a rational expression like \( \frac{x^2 + 2x - 8}{x^2 - 4} \), you need to:

• Factor both the numerator and the denominator as completely as possible. In our exercise, we had \((x - 2)(x + 4)\) in the numerator and \((x - 2)(x + 2)\) in the denominator.
• Cancel out any common factors (as long as they are not in places where the function is undefined, such as \(x = 2\) in this instance).

After canceling the common \((x - 2)\) factor for \(x eq 2\), the expression simplifies to \( \frac{x + 4}{x + 2} \). This simplification allows us to directly evaluate the limit by substituting \(x = 2\) into the simplified expression.
Knowing how to simplify rational expressions allows you to resolve potential issues such as indeterminate forms and analyze limits more easily. It transforms an undefined expression at a certain point into one that's continuous and real at neighboring values.