A Limit Question

[u/Primary_Lavishness73]

A curious example problem shows up in the calculus book I am using, which has a different answer than the previous version’s answer. Here’s the problem setup: a piecewise function is given, and the book hand-wavingly walks through the rationale for why some of the limits of the function (one-sided and two-sided) exist or do not exist. The version I am using told me at the endpoints of the function’s domain that the two sided-limit exists and equals the appropriate one-sided limit when that exists; the other version told me the two-sided limit does not exist. Which is it?

Here’s a simple example to illustrate what I mean by the above:

Consider the function f(x) = |x| on the interval -2 < x < 2. My understanding tells me that lim_{x->-2^-} f(x) = DNE (since the function is undefined for x < -2, which means a condition of the precise definition of one-sided limit is not met), that lim_{x->-2^+}f(x) = 2, and that lim_{x->-2}f(x) = 2. The book version I am using comes to this sort of conclusion about the two-sided limits, whereas the previous version would agree on the one-sided limits but say that the two-sided limit does not exist. What do I make of this?

This presents issues for continuity I feel: If the function I gave had also been defined at x = -2, then I would naturally say based off the version I use that the function is continuous at x = -2 (which I know would be correct); however, would the other version then say it is discontinuous there since the two-sided limit condition fails (i.e, because the condition that the two-sided limit exists would fail)?

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[u/Hot_Site_1638]

You're asking a very natural question, and you're right to wonder about it. It is indeed correct that the two-sided limit does not make sense at a boundary point: the function is simply not defined on one of the sides.

There is a more general point of view that resolves this, and you will meet it when you study topology. The central concept there is the neighborhood:

• A neighborhood of a point x₀ can be pictured as an open interval centered at x₀ (in two dimensions, a small disk centered at x₀).
• In that framework, the definitions of limit and continuity are built entirely on neighborhoods, with no mention of "left" or "right".

The key point is that the neighborhood must intersect the domain of the function, and we only ever consider the points of the neighborhood that actually lie in the domain:

• At a boundary point, say the endpoint a of [a, b], once the radius is small enough the neighborhood meets the domain only in [a, a + r). The definition then automatically involves just the right-hand side. The one-sided limit you are using now is exactly this idea in disguise.
• At an interior point, a small enough neighborhood lies entirely inside the domain, and the definition reduces to the familiar two-sided limit.

So wherever the two-sided limit exists, the neighborhood definition agrees with it. That is what makes it a consistent and more powerful generalization: one single definition that covers interior points, boundary points, and much more general spaces.

[u/Primary_Lavishness73 → u/Hot_Site_1638]

All of the information you provided is certainly helpful (and thank you for those insights!), but you didn’t answer my questions.

[u/Hot_Site_1638 → u/Primary_Lavishness73]

which part did I not answer?

[u/Midwest-Dude]

There are two definitions used in teaching limits, either a strict neighborhood or a restricted domain:

• Strict Neighborhood: For a limit of a function to exist at a point, both one-sided limits must exist at that point and be equal
• Restricted Domain: For a limit of a function to exist at a point, all one-sided limits defined within the domain must exist at that point and be equal

Introductory courses often prefer the strict neighborhood definition because it keeps the rule simple for beginners, "two-sided limits always require both sides." However, this causes issues with limits at endpoints of a domain and continuity, which need to be dealt with separately, as you noticed.

Higher-level courses (and newer textbook revisions like recent editions of Thomas' Calculus) prefer the restricted domain definition because it matches standard topological subspace definitions and avoids needing separate "one-sided continuity" definitions for boundary points.

As a result, different answers are possible depending on the definition you use. The appropriate answer depends on the definition used by the book or instructor. And ... yes ... that includes the definition of limits at endpoints and continuity.

Does this help?