Finding lim sup and lim inf Given the sequence
$$ (a_n)=\begin{cases} 3^{-n}, & \text{for even }n \\ 5^{-n}, & \text{for odd } n \end{cases} $$
How to find:
$$\limsup_{n\to\infty}\frac{a_{n+1}}{a_n}$$
$$\liminf_{n\to\infty}\frac{a_{n+1}}{a_n}$$
$\dfrac{a_{n+1}}{a_n}$ goes like this:
$$\frac{5^1}{3^2},\frac{3^2}{5^3},\frac{5^3}{3^4},\frac{3^4}{5^5},\frac{5^5}{3^6}$$
How can I find lim sup and lim inf?

HINT: First write out a general expression for $\dfrac{a_{n+1}}{a_n}$::
$$\begin{align*}
\frac{a_{n+1}}{a_n}&=\begin{cases}
\dfrac{3^n}{5^{n+1}},&\text{if }n\text{ is even}\\
\dfrac{5^n}{3^{n+1}},&\text{if }n\text{ is odd}
\end{cases}\\\\
&=\begin{cases}
\dfrac15\left(\dfrac35\right)^n,&\text{if }n\text{ is even}\\
\dfrac13\left(\dfrac53\right)^n,&\text{if }n\text{ is odd}\;.
\end{cases}
\end{align*}$$
How do the sequences
$$\left\langle\frac{a_{2n+1}}{a_{2n}}:n\in\Bbb N\right\rangle$$
and
$$\left\langle\frac{a_{2n+2}}{a_{2n+1}}:n\in\Bbb N\right\rangle$$
behave?
