Find the limit.

Question

Find the limit.

lim t→0 tan(15t)

sin(3t)

Answer 1

lim_t→0 [tan(15t) / sin(3t)] = lim_t→0{ [(sin(15t) / cos(15t)] [ 1 / sin(3t)]}

Divide sin(15t) by 15t and multiply 1/sin(3t) by 3t (i.e., multiply the entire expression by (3t)/(15t) = 1/5, then multiply by 5 in front of the limit to compensate). The resulting limit is equivalent to the original and is equal to:

5lim_t→0 [(sin(15t) / (15t))(1/cos(15t))((3t) / sin(3t)) = 5(1)(1)(1) = 5

Recall that lim_θ→0 [sinθ/θ] = lim_θ→0 [θ/sinθ] = 1