real analysis - relations between the root test and the ratio test

relations between the root test and the ratio test

I know the theorem is correct if they are exist

$$\lim\inf\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n} \leq \lim\inf\limits_{n\rightarrow \infty} (A_n)^{1/n} \leq \lim\sup\limits_{n\rightarrow \infty} (A_n)^{1/n} \leq \lim\sup\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n}$$

Here is the 1st question.

If

$$\lim\inf\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n}$$ and
$$\lim\sup\limits_{n\rightarrow \infty} \frac{A_{n+1}}{A_n}$$

are $\infty$

then,

$$\lim\inf\limits_{n\rightarrow \infty} (A_n)^{1/n}$$
and
$$\lim\sup\limits_{n\rightarrow \infty} (A_n)^{1/n}$$

are $\infty$?

And 2nd question is

$$\lim_{n\rightarrow \infty} \frac{|A_{n+1}|}{|A_n|} = \infty$$
then $\lim_{n\rightarrow \infty} (A_n)^{1/n} = \infty$

Actually, 2nd question looks like easy, but I can't prove yet.

Could you please help me?

Thanks