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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- May 3rd 2008, 02:27 PMdamani999Limits 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 - May 3rd 2008, 02:39 PMmr fantastic
I'll give some help with the first bit (needed for the second bit) and let you have another try at the second bit:

$\displaystyle \lim_{n \rightarrow \infty} \frac{n^4 + 1}{n^4 - 4} = \lim_{n \rightarrow \infty} \frac{1 + \frac{1}{n^4}}{1 - \frac{4}{n^4}} = \frac{1 + 0}{1 - 0} = 1$. - May 3rd 2008, 02:45 PMdamani999limits and sequences
is it because 1/n^4 converges to 0 that the L=1. what do u do with this limit?

- May 3rd 2008, 02:57 PMmr fantastic
- May 3rd 2008, 03:22 PMdamani999
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? - May 3rd 2008, 03:37 PMPlato
- May 3rd 2008, 03:44 PMmr fantastic
- May 3rd 2008, 03:51 PMmr fantastic
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.

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

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