anonymous
  • anonymous
Find the values of a and b so that the following is true.
Mathematics
jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
I am thinking about L'Hospital's rule but I am not sure how I would apply it...
anonymous
  • anonymous
well, you can separate them, then solve them
anonymous
  • anonymous
Yeah.... But I have two variables then.

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anonymous
  • anonymous
are you sure it tends to infinity?
anonymous
  • anonymous
Yep.
anonymous
  • anonymous
0.
anonymous
  • anonymous
oops.
anonymous
  • anonymous
why two variables?
anonymous
  • anonymous
if you differentiate the variable you get nothing from a, but b will still be present, you can get b right?
anonymous
  • anonymous
\[\lim_{x \rightarrow 0} (\frac{ \sin(2x) }{ x^3 }+a+\frac{ b }{ x^2 })=0\]
anonymous
  • anonymous
Yeah. I think you are right.
anonymous
  • anonymous
ahhh, so it tends to zero?
anonymous
  • anonymous
Yep :P . I typed wrong.
anonymous
  • anonymous
no problem, you can separate them then do the limits, you can spot it by inspection, very easily
anonymous
  • anonymous
Hmm Let me try...
anonymous
  • anonymous
have you done it?
anonymous
  • anonymous
I can't solve the limit of: \[\lim_{x \rightarrow 0}\frac{ \sin(2x) }{ x^3 }\]
anonymous
  • anonymous
you can!! differentiate 3 times until the denominator is 1,
anonymous
  • anonymous
not 1, differentiate until the denominator is a constant
anonymous
  • anonymous
sin(2)=2sin(x)cos(x)
anonymous
  • anonymous
I got -4/3 .
anonymous
  • anonymous
for?
anonymous
  • anonymous
The limit.
anonymous
  • anonymous
sin(2x)/x^3 as x tends to infinity?
anonymous
  • anonymous
how ?
anonymous
  • anonymous
To 0 :P .
anonymous
  • anonymous
lol!!! im losing it today...
anonymous
  • anonymous
but still, i would have thought it was zero
anonymous
  • anonymous
Waitt.
anonymous
  • anonymous
The Limit is infinity..... :( .
anonymous
  • anonymous
yep,
anonymous
  • anonymous
We can't solve anything then.
anonymous
  • anonymous
we could, i mean if one of the variables was infinity
anonymous
  • anonymous
But We can't do anything with infinity.... It's not a number.
anonymous
  • anonymous
wait are you sure it should tend to 0?
anonymous
  • anonymous
Well I would assume it should tend to a finite value.
anonymous
  • anonymous
i suspect the limit should tend to infinity then for it so satisfy the limit
anonymous
  • anonymous
not necessarily, it can be infinity, let assume it was infinity, then we can do the question with ease. have you done analysis by any chance?
anonymous
  • anonymous
No.....
anonymous
  • anonymous
Here is the question.
1 Attachment
anonymous
  • anonymous
okay, no problem, assume it is infinity because i think you were right the first time round, if it was infinity just plug in infinity to the differential you had computed
anonymous
  • anonymous
If we did that then a would be infinity and b would not exist.
anonymous
  • anonymous
b would, it would be x^2,
anonymous
  • anonymous
one moment,
anonymous
  • anonymous
but since b^2 tends to infinity the whole thing would be 0.
anonymous
  • anonymous
x^2 sorry not b^2.
anonymous
  • anonymous
which would be 0, at when b/x^2 as x ->0
anonymous
  • anonymous
Exactly. We can't work with that.
anonymous
  • anonymous
we could, if b = 0,
anonymous
  • anonymous
0/anynumber = 0
anonymous
  • anonymous
But it's not. B/x^2 is zero, not b.
anonymous
  • anonymous
We don't know what b is.
anonymous
  • anonymous
a could be -infinity, then this would satisfy the limit
anonymous
  • anonymous
But infinity- infinity is not 0.
anonymous
  • anonymous
yes it is
anonymous
  • anonymous
No it's not. Infinity is not a number. It's a concept.
anonymous
  • anonymous
that is true, i moment, i am trying to multi task, i jst remembered the proof that it isnt,
anonymous
  • anonymous
http://en.wikibooks.org/wiki/Calculus/Infinite_Limits/Infinity_is_not_a_number
anonymous
  • anonymous
okay, you need to apply l'hpitals rule over and over again, i would recon, but first you need to put all the equation under one common factor so \[=\frac{\sin(2x) + ax^{3} + bx}{x^{3}}\]
anonymous
  • anonymous
http://answers.yahoo.com/question/index?qid=20110406112720AARpKn1 here is a better and easier way of doing it the sint is taylor expansion i think, the rest is pretty self explanatory
anonymous
  • anonymous
Lol. THanks :P .

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