Determine the following limits.


a. lim x→−2^+ (x − 4) / x(x + 2)

Verified step by step guidance

1. Step 1: Identify the type of limit problem. This is a one-sided limit as \( x \) approaches \(-2\) from the right (denoted as \( x \to -2^+ \)).
2. Step 2: Analyze the function \( \frac{x - 4}{x(x + 2)} \). Notice that the denominator \( x(x + 2) \) becomes zero when \( x = -2 \), which suggests a potential vertical asymptote at \( x = -2 \).
3. Step 3: Consider the behavior of the numerator and denominator as \( x \to -2^+ \). The numerator \( x - 4 \) approaches \(-6\) as \( x \to -2^+ \).
4. Step 4: Examine the sign of the denominator \( x(x + 2) \) as \( x \to -2^+ \). For values of \( x \) slightly greater than \(-2\), \( x + 2 \) is positive, and \( x \) is negative, making the product \( x(x + 2) \) negative.
5. Step 5: Conclude the behavior of the limit. Since the numerator approaches a negative value and the denominator approaches a negative value, the overall expression approaches a positive value. Therefore, the limit is positive infinity.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points of discontinuity or infinity. Evaluating limits is essential for defining derivatives and integrals, which are core components of calculus.

One-Sided Limits

One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side only, either the left (denoted as x→c−) or the right (denoted as x→c+). In the given question, the limit as x approaches -2 from the right (−2+) is crucial for determining the behavior of the function near that point, especially when the function may not be defined at that point.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They can exhibit unique behaviors, such as asymptotes and discontinuities, depending on the values of the variables involved. Understanding how to simplify and analyze rational functions is key to evaluating limits, particularly when the limit involves division by zero or indeterminate forms.