Lim t->0 (sin3tcot5t)/(tcot4t)

[odmart01]

lim t-->0 (sin3tcot5t)/(tcot4t)

Homework Statement

lim t-->0 (sin3tcot5t)/(tcot4t)
I need to find the limit as t approaches 0.

Homework Equations

lim theta-->0 sin(theta)/(theta) =1

The Attempt at a Solution

My attempt is posted, but I'm not sure if its 0 or 12t/5

 

[HallsofIvy]

You have a "t" in the result and that can't be right! You are taking the limit as t goes to 0.

$$\frac{sin(3t)cot(5t)}{tcot(4t)}= 3\frac{sin(3t)}{3t}\frac{cos(5t)}{sin(5t)}\frac{sin(4t)}{cos(4t)}$$

The cosines are no problem, of course.
$$= 3\frac{sin(3t)}{3t}\frac{1}{5}\frac{5t}{sin(5t)}4\frac{sin(4t)}{4t}\frac{cos(5t)}{cos(4t)}$$
The two "t"s that were put into the sine fractions cancel.
Now all of those "sine" fractions go to 1 so the limit is 12/5. There is no "t".