Evaluating problems

Question

Find  .

Options:

1. 5/6
2. 6/5
3. does not exist
4. 0

Answer 1

Recall: lim_θ→0[(sinθ) / θ] = 1 and sinθ is an odd function (so sin(-θ) = -sinθ).

So the given limit can be rewritten as lim_x→0 [sin(6x) / sin(5x)] = lim_x→0 [ (6sin(6x) / (6x)) (5x / (5sin(5x)) ] =

(6/5) lim_x→0 [sin(6x) / (6x)] [lim_x→0 [5x / sin(5x)] = (6/5)(1)(1) = 6/5, since as x→0, 6x→0 and 5x→0.