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Good day, Are there any characteristics of functions such that I can tell on sight which functions have existing limits, and tell them apart from the functions which need further evaluation of left and right hand limits to prove existence of the limit? Then I could save time doing homework or in an exam. Thanks in advance.

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  • $\begingroup$ As starting point, consider : $f(x) = [x : x\geq 0]$ and $[-x: x < 0]$. Intuitively, the slope in the first quadrant is +1, and the slope in the 2nd quadrant is -1. Therefore, the function can not be differentiable at $x=0$, because no matter how arbitrarily small you specify a neighborhood around $x=0$, there will be one value of $x$ in this neighborhood such that $f'(x) = -1,$ and one value of $x$ in this neighborhood such that $f'(x) = 1.$ $\endgroup$ Oct 5, 2020 at 11:13
  • $\begingroup$ Extending previous comment, consider the graph of $f'(x)$, which has a discontinuity at $x=0$. As I understand the definition of limits, there is a direct association between this definition and the notion of continuity. $\endgroup$ Oct 5, 2020 at 11:23
  • $\begingroup$ Sorry, I meant "differentiate" as in tell apart from something, not the calculus function. The sentence has been edited as of now. $\endgroup$
    – Robin Ting
    Oct 5, 2020 at 11:49
  • $\begingroup$ Yes, I got that. My point was that re my 1st comment, although the analysis is clear that the derivative can't exist at $x=0$, that might not be immediately obvious. In the 2nd comment, when you observe that $f'(x)$ has a discontinuity at $x=0$, it becomes more obvious. The idea is (assuming that you reduce the question to one involving continuity), sometimes continuity (or its lack) is not immediately obvious. $\endgroup$ Oct 5, 2020 at 11:53

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