Is it possible to write a problem statement as follows:
A function $f$ is defined on $]0,1[$ as $f(x)=x$. Determine $\lim_{x\to 1}f(x)$.
Or should one write always as:
A function $f$ is defined on $]0,1[$ as $f(x)=x$. Determine $\lim_{x\to 1-}f(x)$.
If only one one these is correct, how to teach that the other one is incorrect?
Context is everything. If you are talking about limits from left and right, above and below, then it's a good idea to indicate using a superscript of + or -. On the other hand, generally when you are talking about a limit without reference to the left or right, above or below, then you are talking about a limit from both directions. What's the conceptual difference?
Limits from one side are about the idea of getting closer and closer in an infinite way. This idea is at the heart of the infinitesimal in calculus (and of supreme importance in analysis further on). A good way of thinking about this is that an infinitesimal is a non-zero number that is so small, that's it's not important in a calculation. Consider how when you take a half of a half of a half, and so on, the distance in question becomes so tiny that it might be better just to round in real life.
On the other hand, limits from two sides of a line, the ones without the superscript are about the idea that if you have an infinitesimal on the left and and an infinitesimal on the right, what you really have is continuity. Continuity is an important idea in calculus, because derivatives are really averages in disguise. And to have an average, you need to have a line determined by two secants.
So you understand, two points can be near to each other, or they can be a specific distance apart. The first is called a 'neighborhood', and the latter is called a 'measure'. The limit of a difference quotient (the derivative) calculates a measure and then attributes the measure to the point, justified by nearness. THAT'S at the heart of the derivative, and the most important lesson from calculus, as well as the hardest part to learn since it's hidden behind so much calcluation and symbols.
So, if you have a point, and you take a point infinitesimally to the left and infinitesimally to the right, you run a line through the points; since the distance between the secant line and the center point is also infinitesimally close, you can pretend that the line parallel to the secant line is the same as the one that is tangent to the curve at the point. That is, the two lines have the same slope because we can "round away" the difference given by the infinitesimal. Thus, a derivative is actually an approximation.
So, whether or not you use the +/- (and it's always a good idea to use them in pairs and not drop the +) is a question of whether you are trying to teach the idea of an infinitesimal, the discrete amount "lower delta x", or trying to teach the idea of continuity, which expresses a geometric intuition about curvature.
Once the relation of the infinitesimal and continuity are understood (the points are infinitely near but not measurable), THEN a student is ready to understand the epison-delta definition of continuity which grounds 'geometric curvature' in 'algebraic implication', logically speaking.
