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

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SUMMARY

The limit as t approaches 0 for the expression (sin(3t)cot(5t))/(cot(4t)) is definitively 12/5. The solution involves applying the limit property lim theta-->0 sin(theta)/(theta) = 1, simplifying the expression by substituting sin(3t) and sin(4t) with their respective limits, and canceling terms appropriately. The final result confirms that there is no remaining variable "t" in the limit, leading to a clear numerical answer.

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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
 

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

[tex]\frac{sin(3t)cot(5t)}{tcot(4t)}= 3\frac{sin(3t)}{3t}\frac{cos(5t)}{sin(5t)}\frac{sin(4t)}{cos(4t)}[/tex]

The cosines are no problem, of course.
[tex]= 3\frac{sin(3t)}{3t}\frac{1}{5}\frac{5t}{sin(5t)}4\frac{sin(4t)}{4t}\frac{cos(5t)}{cos(4t)}[/tex]
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".
 

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