Yes, that's correct. If converges, and exists, then it must be zero. Otherwise, stays above some positive value when x is large enough, say , and the integral is greater than the divergent integral .
The assumption that exists is critical. There are functions whose integral converges but have no limit as x goes to infinity. As a hint, consider the function defined on [0,1], when and when . You can pick any number between 0 and 1 to be the value of its integral on [0,1].