As x approaches 0 x / (sin3x)

Question

To solve a limit as x is approaching 0 while the function is x / (sin3x), what is the step by step process...? I've seen many sites online that go through this problem, yet is completely confusing as to how they complete this problem, and don't state any rules that go through this problem, thank you!

Answer 1

I am certain that all of those explanations obtain the answer 1/3.

 

Try this: limit as x goes to zero of 3x / sin 3x; wouldn't it be easy to replace 3x by t so that you are taking limit (as t goes to 0)  t/sin t.

 

Your calculus book says (and demonstrates why) this limit is 1.

 

lim (x→0) 3•x/sin (3x) = 3 lim(x→0) x/sin 3x = 1. Now divide this last equation by 3, justifying my answer.

 

Also, there's always l'Hopital's rule, telling you that this limit is same as limit of 1/3cos3x and that gives 1/3.

 

I believe that you should practice reading all of those proofs, which no doubt are based on legitimate steps, as published.

 

 

Answer 2

Since this is an indeterminate form, I suggest you use L'Hopital's rule.  Are you familiar with it?

 

If lim_x→a f(x)/g(x) is indeterminate, it is equal to lim_x→a f'(x)/g'(x)

 

lim_x→0 x/sin(3x) = lim_x→0 1/[3cos(3x)] = 1/3