Determine the following limits.


c. lim x→−2 (x − 4) / x(x + 2)

Verified step by step guidance

1. Step 1: Identify the limit expression: \( \lim_{{x \to -2}} \frac{{x - 4}}{{x(x + 2)}} \).
2. Step 2: Check if direct substitution of \( x = -2 \) into the expression results in an indeterminate form. Substitute \( x = -2 \) into the numerator and denominator.
3. Step 3: Notice that substituting \( x = -2 \) gives \( \frac{{-2 - 4}}{{-2(-2 + 2)}} = \frac{{-6}}{{0}} \), which is undefined. This suggests a potential vertical asymptote or a removable discontinuity.
4. Step 4: Factor the denominator if possible to simplify the expression. The denominator is already factored as \( x(x + 2) \).
5. Step 5: Since the expression is undefined at \( x = -2 \), analyze the behavior of the function as \( x \) approaches \(-2\) from the left and right to determine the limit, considering the sign of the numerator and denominator.

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. Evaluating limits is essential for determining continuity, derivatives, and integrals.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that can be multiplied together to obtain the original expression. In the context of limits, factoring can simplify complex rational expressions, making it easier to evaluate limits, especially when direct substitution leads to indeterminate forms.

Indeterminate Forms

Indeterminate forms occur when direct substitution in a limit leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. Recognizing these forms is crucial, as they often require additional techniques, such as factoring, rationalizing, or applying L'Hôpital's Rule, to resolve and find the actual limit.