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

Key Concepts

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.