Which of the following statements, if true, can be used to conclude that f(2) exists?i. limx→2f(x) exists.ii. f is continuous at x=2.iii. f is differentiable at x=2.
In order to determine if f(2) exists, we need to consider the three statements provided
In order to determine if f(2) exists, we need to consider the three statements provided.
Statement i: “limx→2f(x) exists.”
If the limit of f(x) as x approaches 2 exists, it does not guarantee that f(2) itself exists. The limit only tells us the behavior of the function as x gets closer to 2, but it does not give us any information about the value of the function at that specific point. Therefore, statement i alone cannot be used to conclude that f(2) exists.
Statement ii: “f is continuous at x=2.”
If a function is continuous at a specific point, such as x=2 in this case, it means that the value of the function at that point exists and is equal to the limit of the function as x approaches that point. In other words, f(2) exists and is equal to limx→2f(x). Therefore, statement ii can be used to conclude that f(2) exists.
Statement iii: “f is differentiable at x=2.”
Differentiability at a point implies the existence of the derivative at that point. While differentiability implies continuity, it does not guarantee the existence of the function value itself at that point. In other words, a function can be differentiable at a point without necessarily having a defined value at that point. Hence, statement iii alone cannot be used to conclude that f(2) exists.
In conclusion, statement ii – “f is continuous at x=2” – is the only statement that can be used to conclude that f(2) exists.
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