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Some question about limits

  1. Jul 17, 2015 #1
    Hi guys...i'm a little naive...i encountered the limit of this function:

    Sin(x^-1) x

    as the x goes to infinity...in order to study it i know that i have to find the Taylor series about the function Sin(t) centered in 0 having defined t=(x^-1)...something called asymptotical expansion of Sin(x^-1). The fact is that i have not found this technique or the theory behind this so called "asymptotical expansion" in any book! So i was asking of somebody can help me about this with some explanation or some material! thank you!
  2. jcsd
  3. Jul 17, 2015 #2


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    Asymptotic expansion is a process where you expand a function about its limit.
    In ## x \sin \frac1x ## with a large x, you have one large term and one small term, so asymptotic expansion is one good way to understand the behavior of the function for large x.
    A quick search pulls up plenty of resources. One that looks reasonably explanatory is http://www.math.ubc.ca/~feldman/m321/asymptotic.pdf.
  4. Jul 17, 2015 #3


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    You should know that [itex]\lim_{\theta\to 0} \frac{sin(\theta)}{\theta}= 0[/itex]. That is the same as saying that, for small [itex]\theta[/itex], [itex]sin(\theta)[/itex] is approximately equal to [itex]\theta[/itex] and the approximation gets better the smaller [itex]\theta[/itex] is. As x goes to infinity, [itex]\frac{1}{x}[/itex] goes to 0 so [itex]\lim_{x\to \infty} x sin(1/x)= \lim_{\theta\to 0}\frac{sin(\theta)}{\theta}= 1[/itex].
  5. Jul 17, 2015 #4
    Ok, thanks for the reply...for example...how can i find the asymptotic series for Ln(1/Sqrt(1 + x)) ?
    Last edited: Jul 17, 2015
  6. Jul 17, 2015 #5


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    I would go to wolframalpha.com and type in "series ln(1/(sqrt(1+x)))".
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