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If f is continuous on (0,[infinity]) and the integral diverges, and then limx→[infinity]​ doesn't equal 0.

a) This statement is true
b) This statement is false

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Final answer:

The statement is false; a divergent integral does not necessarily imply that the limit of the function as x approaches infinity is non-zero. Counterexamples exist, and more conditions are needed for such a conclusion.

Step-by-step explanation:

The statement 'If f is continuous on (0, infinity) and the integral diverges, and then limx→infinity doesn’t equal 0' is false.

This is because if the integral of a continuous function f from a point to infinity diverges, this does not necessarily mean that the limit of f(x) as x approaches infinity is different from zero.

Counterexamples, such as functions that oscillate infinitely often without settling down to any single value, can maintain a limit that is not equal to zero while their integral diverges.

For a correct statement, you would need further conditions on the function.

Additionally, a divergent integral might be due to the behavior of the function at points other than infinity.

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