Determining the Existence of f(2) in Relation to Limit, Continuity, and Differentiability

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 whether f(2) exists, we need to consider the conditions required for the existence of a function value at a specific point

In order to determine whether f(2) exists, we need to consider the conditions required for the existence of a function value at a specific point.

i. limx→2f(x) exists: The existence of the limit limx→2f(x) does not necessarily guarantee that f(2) exists. While the limit shows the behavior of the function as it approaches x=2, it does not necessarily mean that there is a function value at x=2.

ii. f is continuous at x=2: If f is continuous at x=2, then this would imply that f(2) exists. Continuity at a point means that the function does not have any abrupt jumps or holes at that specific point. Therefore, if f is continuous at x=2, we can conclude that f(2) exists.

iii. f is differentiable at x=2: The differentiability of a function at a point implies continuity at that point. If f is differentiable at x=2, then it is also continuous at x=2. Thus, we can conclude that if f is differentiable at x=2, f(2) exists.

To summarize, the statements (ii) and (iii) can be used to conclude that f(2) exists.

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