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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).
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Don't let this one sink , It's a good question ,guys. I can't work it out tho
Numbers are the essence of the Universe
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[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)
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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.
Why did the vector cross the road?
It wanted to be normal.
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I havent really learned about limit, calculus and stuffs , never take them seriously, When I saw this one, I thought of
, too .Numbers are the essence of the Universe
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