Okay, I attempted 3 of these and I don't know how to do the other two. So I hope I can get you guys to check if I am on the right track and teach me how to do the other two. Thanks a lot, a lot, a lot!

Use the airthmetic of limits, standard limits (clearly stated) or appropriate rules (clearly stated) to compute the limit of each sequence $\displaystyle {a_n}$ if it exists. Otherwise explain why the sequence diverges.

(a) $\displaystyle a_n$ = $\displaystyle \frac{log n + 5n^2}{2n^2 + 100}$

I divided the whole thing with $\displaystyle n^2$ and using standard limits I got,

$\displaystyle \frac{\lim\infty{\frac{log n}{n^2}} + 5}{2}$

Then I used l'Hopital's rule to solve the limit for the $\displaystyle \frac{log n}{n^2}$ and got 1/2 and just substituted it back in and my answer is 11/4.

(b) $\displaystyle a_n = \frac{3^n + n!}{100^n + n^7}$

I don't know how to do this one!

(c) $\displaystyle a_n = (2^n + 1)^\frac{1}{n}$

This one is to use sandwich rule right?

(d) $\displaystyle a_n = cos(\frac {\pi n}{3n + 5})$

Used continuity rule and got,

$\displaystyle cos(\frac {lim \pi n}{lim 3n + 5})$

$\displaystyle cos(\frac { \pi lim n}{3 lim n + lim 5})$

divided the whole thing by n

$\displaystyle cos(\frac { \pi lim 1}{3 lim 1 + lim \frac{5}{n}})$

$\displaystyle cos(\frac {\pi}{3})$

= 0.5

(e) $\displaystyle a_n = n tan(\frac{1}{n})$

I don't know how to do this one either.