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.
A function f(x) is said to be continuous at a point x=a if the limit of f(x) as x approaches a is equal to the value of f(a). So, if 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).
However, the existence of limit (i) limx→2f(x) or the differentiability (iii) f is differentiable at x=2 does not necessarily imply that f(2) exists.
For example, in case (i) f(x) can have a hole or jump at x=2, and the limit may still exist without f(2) existing. In case (iii), the function may not be continuous at x=2, hence f(2) doesn’t exist.
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