Evaluate f(3) if lim x→3^− f(x)=5,lim x→3^+ f(x)=6, and f is right-continuous at x=3.

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Understand the concept of right-continuity: A function f is right-continuous at a point x=a if \( \lim_{x \to a^+} f(x) = f(a) \).

Identify the given information: \( \lim_{x \to 3^-} f(x) = 5 \), \( \lim_{x \to 3^+} f(x) = 6 \), and f is right-continuous at \( x=3 \).

Apply the definition of right-continuity at \( x=3 \): Since f is right-continuous at \( x=3 \), we have \( f(3) = \lim_{x \to 3^+} f(x) \).

Substitute the known limit from the right: \( f(3) = 6 \).

Conclude that the value of \( f(3) \) is determined by the right-hand limit, which is 6.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Limits

Limits describe the behavior of a function as the input approaches a certain value. In this case, we have left-hand and right-hand limits as x approaches 3, which are 5 and 6, respectively. Understanding limits is crucial for analyzing the continuity and behavior of functions at specific points.

Right-Continuity

A function is right-continuous at a point if the limit of the function as it approaches that point from the right equals the function's value at that point. In this scenario, since f is right-continuous at x=3, it implies that f(3) must equal the right-hand limit, which is 6.

Piecewise Functions

Piecewise functions are defined by different expressions based on the input value. In this question, the function f has different behaviors approaching from the left and right of x=3, which is a common scenario in piecewise definitions. Understanding how to evaluate such functions is essential for determining their values at specific points.