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TeX-LaTeX StackExchange is a forum where questions are addressed in a factual manner.

Suppose a responder gets the impression that the questioner is frustrated or desperate because s/he can't figure things out.

(The learning curve for TeX/LaTeX is very steep for many beginners. From my own experience I know that, depending on the expectations of those or the pressure exerted by those who demand that you get to grips with (La)TeX as quickly as possible, this can sometimes even be frightening or increase the stress in life situations that are already riddled with stress. The beginning of a university study, combined with a total change of environment and milieu can be such a situation.)

In such a case, is it appropriate to include encouraging/positive motivational remarks into the factual answer to a question, even if these remarks do not provide any relevant information with regard to the question asked?

I am not concerned with excessive life advice, but with remarks such as:

"In your own attempts to approach the problem, you yourself have already come very close to the solution. The only thing missing for completing the solution is a tiny detail, which is mentioned in the last half-sentence of a double-dangerous-bend paragraph of the TeXbook in a way that can only be understood if you have already read the whole book several times, as recommended in the preface."

Or remarks whose subjectivity cannot be doubted - such as how one felt oneself to be a beginner, but that things became clearer with time, e.g.:

"When I first stumbled upon this problem, it was very confusing, and it took me a while to be able to unravel things as follows: ..."

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    I find that very often the factual answer itself already helps to take away the frustration. People are happy somebody took the time to find their mistake and explain it to them, or point them to some other question that can help solve their problem. But adding remarks like your examples also can help probably. When I add something like that it is usually more aimed at packages or features though, like "the documentation of this package is a bit confusing here" or "unlike the name suggests, ragged right means left aligned" for example.
    – Marijn
    Jun 3 at 21:22
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Very good and very careful thinking.

Written facts can be cold, and data doesn't build a community. We are humans and we enjoy a good conversation. And yes, for people coming here with a technical problem , sympathy and encouragement would be great. And when it comes to criticism of code or question shape, an effort to be especially positive would be appreciated. Even more important for beginners here.

So a big yes from me. It won't hurt readers later to see a story around code.

In addition, comments are also an excellent place for welcoming and for encouraging. I can imagine, that there can be motivating remarks as you wrote above in your question, pointing to and followed by an answer.

For example, based on your words:

Comment to Question: "When I first stumbled upon this problem, it was very confusing, and it took me a while to be able to unravel things. I will write an answer below that explains how I solved it." And an answer follows.

Comment to Question: "In your own attempts to approach the problem, you yourself have already come very close to the solution. I think I know how we can solve it, please see my answer below." And an answer follows.

The way the site works, motivates to write an answer very fast. With respect to this, a fast answer can be accompanied by a later comment in the spirit above, so we still can do it. Or, such a comment shows that someone is already working on it, so no urgent need to respond.

The comment approach has the benefit that users can easily respond and it can be a slightly off-topic talk in a comment thread. Experience shows that years later personal comments, such as a welcome message, are flagged as obsolete and deleted to remove some noise in favour of a good solution database.

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