Determine the following limits.


b. lim x→−2^− (x − 4) / x(x + 2)

Verified step by step guidance

1. Step 1: Identify the type of limit. Since the limit is as x approaches -2 from the left (x \to -2^-), we need to consider the behavior of the function as x gets closer to -2 from values less than -2.
2. Step 2: Analyze the function \( \frac{x - 4}{x(x + 2)} \). Notice that the denominator becomes zero when x = -2, which suggests a potential vertical asymptote at x = -2.
3. Step 3: Determine the sign of the function as x approaches -2 from the left. For x slightly less than -2, both x and (x + 2) are negative, making the denominator positive. The numerator (x - 4) is negative, resulting in the entire fraction being negative.
4. Step 4: Consider the magnitude of the function as x approaches -2 from the left. As x gets closer to -2, the denominator approaches zero, causing the magnitude of the fraction to increase without bound.
5. Step 5: Conclude the behavior of the limit. Since the function becomes increasingly negative as x approaches -2 from the left, the limit is negative 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 where they may not be defined. In this case, we are interested in the limit as x approaches -2 from the left, which is denoted as x→−2^−.

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. The notation x→−2^− indicates that we are considering the limit as x approaches -2 from values less than -2. This is crucial for analyzing functions that may have different behaviors when approached from different directions.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In this limit problem, the function (x − 4) / (x(x + 2)) is a rational function. Understanding how to simplify and evaluate limits involving rational functions is essential, especially when determining points of discontinuity or indeterminate forms.