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Math Help - Limits and sequences

  1. #1
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    Limits and sequences

    hi im stuck on a question and dnt knw what the answer is about..
    Xn=(n^4+1)/(n^4-4). for n>2.
    it asks me to determine L=lim n->infinity Xn
    for each positive real number e, determine an integer No such that |Xn-L|<e for all integers n>No
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    Quote Originally Posted by damani999 View Post
    hi im stuck on a question and dnt knw what the answer is about..
    Xn=(n^4+1)/(n^4-4). for n>2.
    it asks me to determine L=lim n->infinity Xn
    for each positive real number e, determine an integer No such that |Xn-L|<e for all integers n>No
    I'll give some help with the first bit (needed for the second bit) and let you have another try at the second bit:

    \lim_{n \rightarrow \infty} \frac{n^4 + 1}{n^4 - 4} = \lim_{n \rightarrow \infty} \frac{1 + \frac{1}{n^4}}{1 - \frac{4}{n^4}} = \frac{1 + 0}{1 - 0} = 1.
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    limits and sequences

    is it because 1/n^4 converges to 0 that the L=1. what do u do with this limit?
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    Quote Originally Posted by damani999 View Post
    is it because 1/n^4 converges to 0 that the L=1. what do u do with this limit?
    Yes. You use it as the value of L in the second part. I'm sure your class notes or textbook will have an example to follow ......?
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    nope.. my lecture notes arent very informant.. sorry bout that.
    the answers are quite confusing. because they get 1+5/(n^4+1). dnt knw where that 5 came from?
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    Quote Originally Posted by damani999 View Post
    the answers are quite confusing. because they get 1+5/(n^4+1). dnt knw where that 5 came from?
    To get the answer in the text use simple "long division".
    Divide n^4 -4 into n^4 +1.
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    Quote Originally Posted by damani999 View Post
    nope.. my lecture notes arent very informant.. sorry bout that.
    the answers are quite confusing. because they get 1+5/(n^4+1). dnt knw where that 5 came from?
    Another way of doing it is to first note that \frac{n^4 + 1}{n^4 - 1} = \frac{(n^4 -1) + 5}{n^4 - 1} = 1 + \frac{5}{n^4 - 1} ......

    You're advised to thoroughly review the algebra that is pre-requisite for these sorts of questions.
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    Quote Originally Posted by damani999 View Post
    nope.. my lecture notes arent very informant.. sorry bout that.
    the answers are quite confusing. because they get 1+5/(n^4+1). dnt knw where that 5 came from?
    I'm sorry but I find it hard to believe that you would be given a question like this and have no example to refer to, either from class notes or textbook.

    |X_n - L| = \left| 1 + \frac{5}{n^4+1} - 1 \right| = \left|\frac{5}{n^4+1}\right| < \epsilon when n > \left( \frac{5}{\epsilon} - 1\right)^{1/4}.

    So choose N_0 = \left( \frac{5}{\epsilon} - 1\right)^{1/4}.
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