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Is this true?

If converges then f(x) → 0

No — it is false.

The counterexample

f with a spike of height n and width at each integer n

The total area is finite but f is unbounded, so it does not tend to 0.

The kind of mistake this is

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this
integration

Others that fail the same way

From Integration › Improper integrals and convergence tests

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