Determining the Limit of a Function as x Approaches 5: Statement Analysis

The function ff is defined for all xx in the interval 4<x<64<x<6. Which of the following statements, if true, implies that limx→5f(x)=17limx→5f(x)=17 ?

To determine which of the following statements, if true, implies that limx→5f(x)=17, let’s first review the definition of a limit

To determine which of the following statements, if true, implies that limx→5f(x)=17, let’s first review the definition of a limit.

The limit of a function f(x) as x approaches a point c, denoted as limx→cf(x), is the value that f(x) approaches as x gets arbitrarily close to c.

In this case, we are given that the function f(x) is defined for all x in the interval 4 < x < 6. We need to choose the statement that implies that the limit of f(x) as x approaches 5 is 17.

Statement 1: For all x such that 4 < x < 6, f(x) = 17.
This statement directly implies that f(x) is always equal to 17 in the given interval. Therefore, it implies that the limit of f(x) as x approaches 5 is 17.

Statement 2: For all x such that 4 < x < 6, f(x) is continuous.
This statement tells us that f(x) is continuous in the given interval. Continuous functions don’t necessarily imply a specific value for the limit at a single point. Therefore, this statement does not explicitly imply that limx→5f(x)=17.

Statement 3: For all x such that 4 < x < 5, f(x) = 17.
This statement restricts the interval further, only considering x values between 4 and 5. It does not provide any information about the behavior of f(x) as x approaches 5. Therefore, this statement does not imply limx→5f(x)=17.

Based on the analysis, the only statement that implies limx→5f(x)=17 is Statement 1: For all x such that 4 < x < 6, f(x) = 17.

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