Maths Question ~ Mathematicians please help!

titaniumx3

titaniumx3

titaniumx3

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I would really appreciate some advice concerning this question.

QUESTION:

Show that for any real number n.

limit as x->0 of sin(n*x)/sin(x) = n

Now I'm pretty sure you have to use a well known limit such as the one for sin(x)/x but I'm confused as to how you get it into that form. I fear I'm just missing a very obvious trig identity here.

Any ideas?
 
Degree level or college level?
I've got a bunch of maths text books of different levels literally sitting beside me at the mo.
What's the topic called? Trigo?
 
Its degree level and part of real and complex variable theory. They're basic questions revolving around theorems of the limits of functions. Its just that this one has me stumped! :o
 
I'll give you a hint: look up l'hopital's rule. :)

Or I can give you the answer... but that's not really in the spirit of the forum.
 
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if its degree level they wont want you to do this, but as a physicist i'd take the shortcut.

small angle approx

as x-->0
sinx --> x

hence nx/x=n


edit: and i know that isnt a proper proof
 
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LOL, I just about remember doing L'Hôpital's rule last year but yeah I think thats probably the intended way. As for small angle approximations, that would make sense too since I'm not really being asked to prove anything. I just expecting some sort of subtitution/rearrangement job tbh.

Thanks again!


EDIT: I just realised that you can actually rearrange the formula to give:

sin(nx)/nx * n * 1/(sin(x)/x)

Hence as x->0, you're left with n.
 
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Minto said:
surely the first term is 0/0 in that? l'hopital is needed by that route.
Click to expand...
Not if you assume the result sin(x)/x -> 1 as x->0.

However, with that approach you do need to deal with n=0 as a special case. It's the kind of thing an examiner would pick you up on (though you wouldn't lose many marks for missing it).
 
Minto said:
if its degree level they wont want you to do this, but as a physicist i'd take the shortcut.

small angle approx

as x-->0
sinx --> x

hence nx/x=n


edit: and i know that isnt a proper proof
Click to expand...

Dont you mean sinx -> 0 ?!
 
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