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Dharshi

Member

Registered: 2006-10-31

Posts: 56

Need help in trigonometry

If f(x) = (sin 3x+A sin 2x+ Bsin x)/ x^5 for x not equal to zero is continous at x = 0, find A, B and f(0).

Stanley_Marsh

Member

Registered: 2006-12-13

Posts: 345

Re: Need help in trigonometry

Don't let this one sink ,  It's a good question ,guys. I can't work it out tho

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Numbers are the essence of the Universe

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Need help in trigonometry

[EDIT: No, I got it wrong. Forget what I said previously.]

Anyway …

I suspect that the analytical proof would make use of the following useful result:

Last edited by JaneFairfax (2007-03-15 11:35:28)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Need help in trigonometry

The following may well be a very inefficient method for solving, but it's the only one I can think of.

If f(x) is continuous at x=0, then

exists.

L'Hopital's rule says that if f(x) = g(x) = 0, then

.

Therefore,

.

Now, the denominator of that fraction is clearly still 0, but the numerator is unknown, due to it involving A and B. However, a finite value divided by x will approach infinity as x goes to 0, and that's not a well-defined limit. So as we know that f(x) is continuous, that means that the numerator must be 0.

Therefore, 3+2A+B = 0.

We've established that both sides of the fraction are 0, so now we can use L'Hopital's rule again.
In fact, we can use it again and again until the denominator becomes a constant, at which point the limit will be found.

.

The second line shows an expression whose numerator and denominator must be 0, by similar reasoning to above, and so another equation can be derived: -27 - 8A - B = 0.

We now have two simultaneous equations for A and B, and solving these gives A = -4, B = 5.
Now we can use these and the final line to find f(0).

What a wonderfully simple answer for such a horribly complex method.

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Why did the vector cross the road?
It wanted to be normal.

Stanley_Marsh

Member

Registered: 2006-12-13

Posts: 345

Re: Need help in trigonometry

I havent really learned about limit, calculus and stuffs , never take them seriously, When I saw this one, I thought of

, too .
  mathsyperson's proof exccedes my knowledge lol , But I think I've got the idea.lol

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Numbers are the essence of the Universe