How can we realize when a sociological question is impossible to answer? In my language there is a famous common sense statement which tells the populations are swinging back and forth from one extreme to the other. 

The idea behind is that, at a sociological level, the influence of many psychological factors (such as the dichotomous thinking and similar cognitive distortions) are leading the  populations to tend to big breaks and changements. 

Would it be possible to investigate it from a sociological point of view? How can we realize when a sociological question is impossible to answer?
Short version: A question is possible to answer sociologically if and only if its scale and region (informally speaking, its context) are completely defined and its variables of interest are operationalized.
Long (really long) version: The following answer may be colored by my quantitative background, but if you give me the benefit of the doubt here and there, this should generalize fairly well. My apologies for any untoward implications in advance.
Is my question amenable to empirical inquiry?
This is more than a question unto itself, but I will try to keep it as brief and general as possible while still answering the sociology-specific question. Sociological questions must be amenable to empirical inquiry. A question is amenable to empirical inquiry if and only if:

The variables of interest have been operationalized. If we are interested in &quot;society,&quot; how exactly do you measure change in society? (I strongly recommend reading mfloren's excellent answer in the link.)
The region of interest has been completely defined. If we are interested in &quot;society,&quot; where exactly should we start measuring our variables of interest?

There are (very) different views on how to interpret the evidence so collected, a fair covering of which is beyond the scope of this answer. In the cognitive sciences, I would say the dominant views tend to be in line with falsificationism and/or instrumentalism. Broadly speaking, this area is covered by the study of epistemology.
What is the scale of my question?
A question's scale refers to the level of detail involved. How &quot;big&quot; is the question? Informally speaking, the scale of an empirical question conveys information about the characteristics of the spacetime region we are investigating, and what level of precision/detail we are investigating it at.
Example: Cosmological or Quantum?
A prototypical example here that I think most people are familiar with from basic physics would be the cosmological scale—the largest known scale—and its opposite, the quantum scale, which is the smallest known scale. Informally speaking, the &quot;physics scale&quot; runs from least energy involved to most energy involved.
This is the simplest to understand because most of physics can be captured by how much energy is involved. In the biological sciences, it becomes more complicated. We might talk about trophic levels in a conceptually similar way when asking questions about ecologies, but we might also talk about the embryonic or neural scales for other questions. In other words, they are, in a very direct sense, different facets of the same &quot;object.&quot; In general, the closer we get to a first-person human level of analysis, the more difficult it becomes to sufficiently define questions, and the further away we get, the easier it becomes.
Sociobehavioral science: Within or between individuals or groups?
In the social and behavioral sciences, we tend to look at the amount of people involved, the qualitative relationship between those people, and whether we are interested in the question at the intra-individual, between-individual, intra-group and/or inter-group level. As with the cosmological and quantum scale, or embryonic and neural scales, these are not ontologically distinct—scales are a strictly epistemological concept. Informally speaking, a question is sociological if it is a question about the collective behavior of relatively large groups of people (i.e., if regions and variables are defined relative to other 'groups' rather than in terms of their constituents' individual behavior or minds).
What field is appropriate for my question?
Once we have defined the scale and region (or area) of our question, and we have operationalized our variables of interest, we can say whether a particular  field is an appropriate context for situating our question in. If our question is, &quot;How do modern bridges hold up so much weight?&quot;, then we could situate this in terms of quantum physics, but doing so would be a grossly excessive level of detail, so we'd probably want to use engineering instead.
So it is for the study of behavior and the human condition: it is important to use the right tool for the job. (Actually, I prefer to liken fields to painting styles, but the tool metaphor tends to get the job done faster, and I feel like this has already gotten long enough.)