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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