It would definitely be possible to proceed. One simply evaluates the limit. Now that you know the final answer, it should be clear that the first term from the integral (i.e. the T/2) is the part that will give you your final answer, and the other term should go to zero in the limit.
I recommend using the following 2 trig. identities to make it easier.
After you do this, you can simplify and you will end up with a limit of the well known sinc funtion (https://en.wikipedia.org/wiki/Sinc_function), which goes to zero as the argument goes to infinity.