Edited for assignment purposes...Write the KVL differential equation form for the simple series loop comprising the capacitor, inductor and two 60Ω resistors. Use the voltage on 1mF C at t=0 as the initial condition and decide what the current will be at t=∞. You've correctly stated that i(0+)=0.
Then solve the differential equation per standard methods to find i(t).
Edited for assignment purposes...Correct - but I'm tired of holding your hand.
Presumably Wiz0r doesn't want their circuit theory teacher too easily deducing they were receiving help online.This thread will certainly help those who come along in 6 months asking the same question.