$\lim_{n\to\infty}a_nb_n=? $ given that $\lim_{n\to\infty}a_n=0$ and $\lim_{n\to\infty}b_n=\infty$ A. $\lim_{n\to\infty}a_nb_n=?$ given that $\lim_{n\to\infty}a_n=0$ and $ \lim_{n\to\infty}b_n=\infty$ 
B. $\lim_{n\to\infty}{a_n \over b_n}=?$ given that $\lim_{n\to\infty}a_n=0$ and $\lim_{n\to\infty}b_n=0$ 
C. $\lim_{n\to\infty}{a_n \over b_n}=?$ given that $\lim_{n\to\infty}a_n=\infty$ and $\lim_{n\to\infty}b_n= \infty$  

These seem to be difficult situations I come across when I have to evaluate limits. 

Some examples, 


$\lim_{n\to\infty}n^a\sin(1/n^b)$ with $a,b&gt;1$  
$\lim_{n\to\infty}{n^a\over \sin(1/n^b)}$ with $a,b&gt;1$  
$\lim_{n\to\infty}{1/n^a\over \sin(1/n^b)}$ with $a,b&gt;1$  
$\lim_{n\to\infty} nf(x_n)$ [where we only know that $\lim \ f(x_n) = 0]$ 


My first question is : Are the two situations  A and B  equivalent ? I feel that they are equivalent as $\lim_{n\to\infty}b_n=\infty \iff \lim_{n\to\infty}{1\over b_n}=0$, The problem is that I'm not sure if we can apply the product rule for limits to take the limit inside.

My second question is : What are some ways to solve such problems ? I only know that we can apply stolz- cesaro in some cases.
These are usually called indeterminate forms, not because we can't compute the limit, but because no general theorem exists.

Just to make a couple of easy examples for your fourth question.

Consider $b_n=\frac{1}{n}$. Then
$$
\lim_{n\to\infty} b_n=0
$$
and
$$
\lim_{n\to\infty} nb_n=\lim_{n\to\infty}n\frac{1}{n}=1
$$
If we take instead $c_n=\frac{1}{n^2}$ we have
$$
\lim_{n\to\infty} c_n=0,\qquad
\lim_{n\to\infty} nc_n=\lim_{n\to\infty}\frac{1}{n}=0
$$
With $d_n=\frac{1}{\sqrt{n}}$ we have
$$
\lim_{n\to\infty} d_n=0,\qquad
\lim_{n\to\infty} nd_n=\lim_{n\to\infty}\sqrt{n}=\infty
$$

So no general rule can be given.

You may want to check that
$$
\lim_{n\to\infty} n^a\sin\frac{1}{n^b}=
\begin{cases}
\infty &amp; \text{if $a&lt;b$}\\
1 &amp; \text{if $a=b$}\\
0 &amp; \text{if $a&gt;b$}
\end{cases}
$$
For example, let's look at your question B, the ratio of sequences.

What is the answer when we set $a_n=\frac{1}{x}$, $b_n=\frac{1}{x}$?

What about when $b_n=\frac{1}{x^2}$?

When $a_n=\frac{1}{x^2}$ and $b_n=\frac{1}{x}$?