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.
To determine whether f(2) exists, we need to consider if the function f is defined and continuous at x = 2
To determine whether f(2) exists, we need to consider if the function f is defined and continuous at x = 2.
Statement i: limx→2f(x) exists
If the limit of f(x) as x approaches 2 exists, it suggests that there is a defined value for f(2). However, it does not guarantee that the function is continuous at x = 2. So, statement i alone is not sufficient to conclude that f(2) exists.
Statement ii: f is continuous at x = 2
If the function f is continuous at x = 2, it means that the limit of f(x) as x approaches 2 exists and is equal to f(2). So, if statement ii is true, it implies that f(2) exists.
Statement iii: f is differentiable at x = 2
The fact that f is differentiable at x = 2 implies that the function is continuous at x = 2. Therefore, if statement iii is true, it also guarantees that f(2) exists.
In summary, either statement ii or statement iii alone can be used to conclude that f(2) exists.
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