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 conclude that f(2) exists, we need to consider the different conditions that ensure the existence of the function value at x = 2
To conclude that f(2) exists, we need to consider the different conditions that ensure the existence of the function value at x = 2.
i. The fact that the limit of f(x) as x approaches 2 exists does not necessarily guarantee that f(2) exists. The limit only provides information about the behavior of the function as it approaches a particular value, but not its actual value at that point. For example, it is possible for a function to have a limit at a certain point but not be defined at that point.
ii. If f is continuous at x = 2, then we can conclude that f(2) exists. Continuity implies that there are no gaps, holes, or jumps in the function. Therefore, if f is continuous at x = 2, it means that the function is defined and has a value at that point.
iii. If f is differentiable at x = 2, then it implies that the derivative of f with respect to x exists at x = 2. This condition does not guarantee that f(2) exists. Differentiability only ensures the existence of the derivative, which measures the rate of change of the function. It does not guarantee that the function is defined at x = 2.
Therefore, the statement that can be used to conclude that f(2) exists is ii. f is continuous at x = 2.
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