# Limit from both sides or from left? [closed]

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?

• There are various discussions on this in MSE, for example math.stackexchange.com/questions/4198573/…. The definition of limit, especially at higher dimensions, is not the value when both one-sided limit agrees. Limit has to do with the agreement of values when approached from all sides. If you can only approach from one side, then the one-sided limit is the limit. Both forms are correct. Not posting as an answer because I am unsure if this counts as a duplicate question. Commented Dec 16, 2021 at 3:49

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.

• This doesn't answer the question, which is about how the definition of a limit (and one sided limit) interact with the domain of a function. Commented Dec 16, 2021 at 11:14
• Hi. :D Let's note: 1. We might infer the asker of the question disagrees with you based on the confirmation vote. 2. This question doesn't ask about a domain, but about a notation and we can infer this because the variation in (1) and (2) are in the limit subscript, not the notation to express the domain. 3. The actual question presumes that one notation is correct, and the other is incorrect. It is a bad presumption, and I have rebutted it. So, to synthesize these observations, what this question asks is which notation is correct, and how to teach that...
– J D
Commented Dec 16, 2021 at 17:19
• My response is to undermine the premise and commit to the idea that whether or not the notation presented is correct is a matter of the which concept is being taught. (1) is used within the context of teaching 'continuity', but is usually used in textbooks after (2) is taught, because the definition of (1) is built on the (2). Now I'm open to an explication of why you claim this question "interacts with the domain of a function" as I see neither artificial nor natural language that supports the claim. The question asks, "how do I teach a math concept related to a difference in notation"...
– J D
Commented Dec 16, 2021 at 17:23
• @J D I will put my reading of the OP's question into more precise language: Let $f$ be a function defined on the open interval $(0,1)$ (they use the slightly archaic notation $]0,1[$ for this interval). Then, since $f$ is not even defined for values greater than 1 is it acceptable to use $\displaystyle \lim_{x \to 1} f(x)$, since this seems to imply a two sided limit? Or must I instead use $\displaystyle \lim_{x \to 1^{-}} f(x)$, since $x$ can only approach $1$ from the left? Commented Dec 16, 2021 at 17:36
• This is off topic because it is a technicality, and a math question. It is answered in the comment by okzoomer to the original post: the definition of the limit of a function used in most analysis texts would permit either notation in this case. Commented Dec 16, 2021 at 17:38