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Limit value

  1. Apr 20, 2004 #1
    This question appeared in today's exam but I think I did it wrongly. I use the "sum to product formula", but I can't find a limit value and say it doesn't exist. Am I correct?

    [tex]\lim_{n\rightarrow\infty} cox \sqrt{2004 + x} - cos \sqrt{x} [/tex]
     
  2. jcsd
  3. Apr 20, 2004 #2

    arildno

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    I take it that you are to find the limit when x->inf, not n->inf.
    The limit is zero:
    Note that sqrt(a+x)=sqrt(x)*sqrt(1+(a/x)) approx. sqrt(x)+1/2(a/sqrt(x)), when a<<x

    Hence, we may write the original cosine as cos(sqrt(x)+e),
    where e->0 as x->inf.

    Using sum-to-product, we have to evaluate the limit of:
    cos(sqrt(x))*(cos(e)-1)-sin(e)*sin(sqrt(x)).

    Since cos(sqrt(x)), sin(sqrt(x)) are bounded by 1, we see that the whole expression goes to 0.
     
  4. Apr 22, 2004 #3
    I see. Thanks
     
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