Picture shows a dumb approach to finding the limit.
Just expand Tan(6t) and sin(2t) and simplify the ratio to:
Yes, if you "leave out" the "t"s you have hust and but that doesn't help! If you then take the limit as t goes to 0 you have and which leads to - and that doesn't give anything. What they are doing is using the basic limit, with x= 6t in the first case and x= 2t in the second.