A function f is even if f(−x)=f(x), for all x in the domain of f. Suppose f is even, with lim x→2^+ f(x)=5 and lim x→2^− f(x)=8. Evaluate the following limits.


a. lim x→−2^+ f(x)

Verified step by step guidance

1. Identify the property of even functions: f(-x) = f(x) for all x in the domain of f.
2. Recognize that the limit as x approaches a value from the right (x → a^+) is concerned with values slightly greater than a.
3. Since f is even, f(-x) = f(x), so lim x→-2^+ f(x) = lim x→2^- f(x).
4. Use the given information: lim x→2^- f(x) = 8.
5. Conclude that lim x→-2^+ f(x) = 8 based on the even function property.

Key Concepts

Even Functions

An even function is defined by the property that f(−x) = f(x) for all x in its domain. This symmetry about the y-axis implies that the function takes the same value for both positive and negative inputs of the same magnitude. Understanding this property is crucial for evaluating limits involving even functions, as it allows us to relate the behavior of the function at positive and negative values.

One-Sided Limits

One-sided limits refer to the behavior of a function as it approaches a specific point from one side only, either the left (denoted as lim x→c^−) or the right (denoted as lim x→c^+). In this question, the limits as x approaches 2 from the right and left are given, which are essential for understanding the overall limit behavior of the function at that point. Evaluating one-sided limits helps in determining continuity and the existence of limits at specific points.

Limit Evaluation

Limit evaluation involves determining the value that a function approaches as the input approaches a certain point. In this case, we need to evaluate lim x→−2^+ f(x) using the properties of even functions and the provided one-sided limits. Recognizing that f is even allows us to infer that f(−2) will equal f(2), thus connecting the limits at positive and negative values to find the desired limit.