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
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:
$\displaystyle \lim_{n \rightarrow \infty} \frac{n^4 + 1}{n^4 - 4} = \lim_{n \rightarrow \infty} \frac{1 + \frac{1}{n^4}}{1 - \frac{4}{n^4}} = \frac{1 + 0}{1 - 0} = 1$.
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 $\displaystyle N_0 = \left( \frac{5}{\epsilon} - 1\right)^{1/4}$.