Question1:

Can you see some similarity bewteen infinitely large sequences and infinitely small squences?

Answer:

In case of infinitely large sequences any term of the sequence can't reach $\infty$ hence we say limit doesn't exist right? No, you are wrong. This is because the $\infty \notin \mathbb{R}$.
Moreprecisely, any sequence in this limit world can't reach to its limit that's why the name limit.

Similarly, in case of infinitely small sequence the name infinity comes and it actuallly says each term $a_n$ tries to go near 0 using its infinite terms. Is not it so cool?

What we learn now? For infinitely large sequence limit doesn't exists as $\infty \notin \mathbb{R}$. But as $0 \in \mathbb{R}$ for infinitely small sequence limit exists. Then we can think of a set where $0 \notin$ the set, and hence if a sequence from that set coverge or gets closer to $0$ we will say limit of that sequence doesn't exists although it is actually converging or getting closer to $0$. A example of such a set is $\mathbb{R}\setminus \{0\}$. We will discuss it later why this is hapening so. It is actually related to something where a order set doesn't have a GLB or LUB kind of thing. Whatever leave it for now. We will connect it later.

Main Idea: The main idea was to make you think that we can actually add the zero to the set $\mathbb{R}\setminus \{0\}$. And then our infinitely small sequence gain start converging to $0$. Similarly, if we add $\infty$ to $\mathbb{R}$ then can we say our infinitely large sequence is converging to $\infty$ in the set $\mathbb{R} \cup \{\infty\}$?
Answer looks to be positive quite promisingly right? And and our feelings are correct. The answer is yes and this kind of mathematical territory is called extended real number system.

Congratulations. You learned a very new, intersting and deep concept of mathematics. And yes, if you think is this how simple? And the answer is yes. Mathematics is this much simple like water if you know the picture under the water and can map all its depth. Unless you will be lost in dark deep see and can never come out of it.
Mathematics looks like it is a see with unknown depth and yes it is. Looking from the top will make you think it is beautiful. Going in sallow water will make you feel uncomfortable and finally it will swallow you with deep dark depth if you can't picture the depth of it. Just try to think with pictures. The sea is full of resources and beauty for you and you will get them more simply than no one can get.