Can someone help me with a good strategy for finding the limit of a sequence? I am just not understanding the best way to solve these. What I have been trying to do to first take the limit. If i got an indeterminate, I changed it to a function and took the derivative, but I'm not sure that is the way I need to evaluate all of these. Any help is greatly appreciated!

1) $\displaystyle a_n = 1 - (0.1)^n

$

$\displaystyle \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$

2) $\displaystyle a_n = \frac{n^3}{3n + 1}$

$\displaystyle \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$

3) $\displaystyle a_n = e^{7/(n+9)}$

$\displaystyle

\lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$

4) $\displaystyle a_n = \frac{3^{n+1}}{5^n}$

$\displaystyle \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$

5) $\displaystyle a_n = tan (\frac{6n\pi}{4 + 24n})$

$\displaystyle \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$

6) $\displaystyle a_n = n^2e^{-5n}$

$\displaystyle \lim_{n\rightarrow \infty}$ $\displaystyle a_n = ?$