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 determine if there are any conditions that ensure the existence of the value of f at x = 2
To conclude that f(2) exists, we need to determine if there are any conditions that ensure the existence of the value of f at x = 2.
i. limx→2 f(x) exists: This statement alone may not guarantee the existence of f(2). The existence of f(2) depends on the behavior of the function as x approaches 2, not just its limit.
ii. f is continuous at x=2: This statement does guarantee that f(2) exists. Continuous functions are defined at every point in their domain, including the specific point x = 2. Therefore, if f is continuous at x = 2, it means that f(2) exists.
iii. f is differentiable at x=2: This statement also guarantees that f(2) exists. Differentiability implies continuity, so when a function is differentiable at a specific point x = 2, it is also continuous at that point. As mentioned earlier, if f is continuous at x = 2, it means that f(2) exists.
In conclusion, both statement ii (f is continuous at x=2) and statement iii (f is differentiable at x=2) can be used to conclude that f(2) exists. Statement i (limx→2 f(x) exists) alone does not guarantee the existence of f(2).
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