Push it to the limit?

Question

lim tan3t
t-->0 cos4t.sin4t

Answer 1

Your question is not so clear to me. What is your question?
Is that lim (t-->0) tan 3t/(cos 4t.sin 4t)?

lim(t-->0) tan 3t / (cos 4t.sin 4t)
=lim(t-->0) sin 3t / (cos 3t.cos 4t.sin 4t)
=lim(t-->0) (sin 3t/sin 4t).lim(t-->0) 1 / (cos 3t.cos 4t)
=lim(t-->0) (sin 3t/3t)/(sin 4t/4t).(3/4).lim(t-->0) 1 / (cos 3t.cos 4t)
=(1/1).(3/4).(1/1)
=3/4

Answer 2

Do you mean

lim (t->0) (tan 3t) / (cos 4t * sin 4t) ?

I'll assume you do. Then you can rewrite it as

lim (t->0) (sin 3t) / (cos 3t*cos 4t*sin 4t).

lim (t->0) 1/(cos 3t*cos 4t) = 1/(cos 0*cos 0) = 1/(1*1) = 1, so we just focus on

lim (t->0) sin 3t / sin 4t.

As t->0, sin nt -> 0 for any n, so we use l'Hospital's Rule:

lim (t->0) 3 cos 3t / (4 cos 4t) = 3/4

because lim (t->0) cos nt = 1 for any n.