Does $\lim \frac {a_n} {b_n}$ exist and $\lim a_n \neq 0$ imply $\lim b_n$ exist? Suppose $\lim_{n \rightarrow \infty} \frac {a_n} {b_n}$ exist and $(a_n)$ converges to some number $k \neq 0$. Is it then possible to conclude that $(b_n)$ converges ? 
Also, suppose $\lim_{n \rightarrow \infty} \frac {a_n} {b_n}$ exist and $(b_n)$ converges to some number $k \neq 0$. Is it then possible to conclude that $(a_n)$ converges ? 
I am well aware of the statement that if $(a_n)$, $(b_n)$ converges and $b_n\neq 0$ for $n \ge N$ then $\lim_{n \rightarrow \infty} \frac {a_n} {b_n} = \frac {\lim_{n \rightarrow \infty} a_n} {\lim_{n \rightarrow \infty} b_n}$. I've tried to use the contrapositive of this statement to prove my hypotheses. It should be said, that this is not an exercise, but something I've been wondering about.

For the first statement you might try to do:$\lim\frac{a_n}{\frac{a_n}{b_n}}$ but if you consider the case where: $a_n=1+\frac 1n$ and $b_n=n$ then $\lim\frac{a_n}{b_n}=\lim \left( \frac 1n+\frac 1{n^2}\right)=0$ and $\lim a_n=1$ but $\lim b_n=\infty$
And for the second part:$\frac{a_n}{b_n}b_n=a_n$
$\frac{a_n}{b_n}$ is converges and $b_n$ is converges and product of two converges series is converges therefore $a_n$ is converges
