Use the graph of f(x) = x / (x² − 2x − 3)² to determine lim x→−1 f(x) and lim x→3 f(x).

Verified step by step guidance

1. Identify the function: \( f(x) = \frac{x}{(x^2 - 2x - 3)^2} \).
2. Factor the quadratic in the denominator: \( x^2 - 2x - 3 = (x - 3)(x + 1) \).
3. Rewrite the function: \( f(x) = \frac{x}{((x - 3)(x + 1))^2} \).
4. Analyze the behavior of \( f(x) \) as \( x \to -1 \): The denominator becomes zero, indicating a potential vertical asymptote or removable discontinuity.
5. Analyze the behavior of \( f(x) \) as \( x \to 3 \): Similarly, the denominator becomes zero, indicating another potential vertical asymptote or removable discontinuity.

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 the function's value at points where it may not be explicitly defined, such as points of discontinuity or asymptotes. Evaluating limits often involves techniques like substitution, factoring, or applying L'Hôpital's rule.

Continuity

Continuity refers to a property of a function where it is uninterrupted and has no breaks, jumps, or holes in its graph. A function is continuous at a point if the limit as the input approaches that point equals the function's value at that point. Understanding continuity is essential for evaluating limits, especially when determining if a limit exists at a specific point.

Asymptotes

Asymptotes are lines that a graph approaches but never touches or crosses. They can be vertical, horizontal, or oblique and indicate the behavior of a function as it approaches certain values. Identifying asymptotes is crucial for understanding the limits of a function, particularly in cases where the function may tend toward infinity or exhibit undefined behavior at specific points.