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.
II and III onlyIf ff is continuous at x=2, then f(2) exists. Also if f is differentiable at x=2, then ff is continuous at x=2 and f(2) exists.
The statement that can be used to conclude that f(2) exists is (ii) f is continuous at x=2.
Explanation:
(i) The existence of the limit limx→2f(x) doesn’t necessarily mean that f(2) exists. For example, the function f(x) = 1/x has a limit of 0 as x approaches 2, but its value at x=2 doesn’t exist.
(ii) If a function is continuous at a point, it means that the limit of the function at that point exists and is equal to the value of the function at that point. Therefore, if f is continuous at x=2, then f(2) exists.
(iii) If a function is differentiable at a point, it means that the limit of the difference quotient exists at that point. However, the existence of the limit of the difference quotient doesn’t imply the existence of the function value at that point.
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