The limit...
(1)
... doesn't exist but exists the limit...
(2)
Welll...yes, in a rather trivial way: since INT{0,t} sin(s) ds = -cos(t) + 1,
then lim (1/t)[1 - cos(t)] = 0 when t --> oo since 1 - cos t is bounded and (1/t) --> 0 .
This though doesn't seem to appropiately address the OP's question.
Tonio
In general if
is the sum of a finite number of periodic functions
not necessarly with the same period the limit...
(3)
... exists and
is called [improperly] 'mean value of
'. It is...
(4)
Kind regards