Limits of trig functions: lim (sin3x cot 5x) xappr___________ (x cot4x)

has_major_problems

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I am trying to solve this problem but I keep getting a answer I know is wrong. I know the answer, I just need a walk through of the steps. I am currently banging my head against a wall trying to figure this out.
lim (sin3x cot 5x)
xappr___________
(x cot4x)


I know the answer 12/5, I just can't get there.
 
… I keep getting a answer I know is wrong …

xappr
That is meant as x->0, yes?

May we see what you tried? At this point, we can't know whether you've made a single misstep, or whether the approach is off or what else might have happened along the way. It would also help to know what you've studied. Have you covered derivatives, yet? I want to apply l'Hopital's rule on sin(3x)/x, but if you're just beginning calculus that's not an option. (Direct substitution works on the rest of the given limit.)
 
I am trying to solve this problem but I keep getting a answer I know is wrong. I know the answer, I just need a walk through of the steps. I am currently banging my head against a wall trying to figure this out.
lim (sin3x cot 5x)
xappr___________
(x cot4x)


I know the answer 12/5, I just can't get there.

Here is a similar example. To find the limit as x->0 of sin(3x)/sin(7x), we can use the known fact that the limit of sin(x)/x is 1, by rewriting the expression as sin(3x)/(3x) * (7x)/sin(7x) * (7x)/(3x), where I have inserted 3x and 7x each once on top and once on the bottom in order not to change the value, but pairing terms up so that the three factors have known limits, namely 1, 1, and 7/3. So the limit is 7/3.

Does that look like something you have been taught? There are several ways to present it, so if you don't follow this, you should show us (1) your own attempt, showing us what kind of mistake you are making, and/or (2) an example from your textbook, showing how they are teaching it.
 
I am trying to solve this problem but I keep getting a answer I know is wrong. I know the answer, I just need a walk through of the steps. I am currently banging my head against a wall trying to figure this out.
lim (sin3x cot 5x)
xappr___________
(x cot4x)


I know the answer 12/5, I just can't get there.
You should write everything in terms of sines and cosines, separate the sines and cosines and then get the same angle above or below what is in sine function. Then use the fact that sin(kx)/(kx) and kx/sin(kx) both approach 1 as x approaches 0
 
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