The reason f′(0) is undefined for the absolute value function is that you need the value to be the same for all sequences converging to 0 – both from the left and from the right. There’s a nice way to motivate this in higher-dimensional settings by thinking about the action of e.g. complex multiplication, but this is a much stronger notion than real differentiability and I’m not quite sure how to think about motivating the single-valued real case yet. Of course, you can say things like “the theorems just work out nicer if you require both the lower and upper limits be the same”...
The reason f′(0) is undefined for the absolute value function is that you need the value to be the same for all sequences converging to 0 – both from the left and from the right. There’s a nice way to motivate this in higher-dimensional settings by thinking about the action of e.g. complex multiplication, but this is a much stronger notion than real differentiability and I’m not quite sure how to think about motivating the single-valued real case yet. Of course, you can say things like “the theorems just work out nicer if you require both the lower and upper limits be the same”...