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.
lim x→−2^− f(x)

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Identify the property of even functions: An even function satisfies the condition f(-x) = f(x) for all x in its domain.

Given that f is even, we have f(-x) = f(x). Therefore, f(-2) = f(2).

Examine the given limits: \( \lim_{x \to 2^+} f(x) = 5 \) and \( \lim_{x \to 2^-} f(x) = 8 \).

Since f is even, the behavior of f around x = -2 will mirror the behavior around x = 2. Thus, \( \lim_{x \to -2^-} f(x) = \lim_{x \to 2^+} f(x) = 5 \).

Conclude that \( \lim_{x \to -2^-} f(x) = 5 \) based on the symmetry of even functions.

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주요 개념

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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 and determining function behavior around specific points.

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 (−) or the right (+). In this case, lim x→2^+ f(x) and lim x→2^− f(x) indicate the values of the function as x approaches 2 from the right and left, respectively. These limits help in understanding discontinuities and the overall behavior of the function near critical points.

Limit Evaluation at Symmetric Points

When dealing with even functions, the limits at symmetric points can be inferred from each other. For instance, if f is even and we know lim x→2^− f(x) = 8, we can deduce that lim x→−2^− f(x) will also equal 8, since f(−2) must equal f(2) due to the even property. This relationship simplifies the evaluation of limits for even functions.

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.lim x→−2^− f(x)