Section 6.3 Limits of the form nonzero over zero: Enlarging the definition of limit

[markbarsamian]

Section 6.3 Limits of the form nonzero over zero: Enlarging the definition of limit

My second comment about this section is more substantive. I think it is very important to point out that in this section, we are essentially enlarging the definition of limit.

In Section 4.2 What is a Limit, the value of a limit had to be a number. For example, if we were to compute the limit, as x approaches 3 from the left, of f(x) = (x-7)/(x-3), the answer would be that the limit does not exist because the section 4.2 limit laws do not allow us to find the limits of the form nonzero over zero.

But now, in Section 6.3, we say that that same limit does exist: the limit is infinity.

But there is more subtlety here that should be discussed. Using the Section 4.2 limit laws, if we consider the two-sided limit, that is, if we were to compute the limit, as x approaches 3, of f(x) = (x-7)/(x-3), the answer would again be that the limit does not exist because the section 4.2 limit laws do not allow us to find the limits of the form nonzero over zero.

And using Section 6.3 techniques, if we were to compute the limit, as x approaches 3, of f(x) = (x-7)/(x-3), the answer would also be that the limit does not exist. But the limit does not exist for a different reason. Using Section 6.3 techniques, the limit does not exist because the left and right limits don't match.

(Note that nowhere in Section 6.3 do you discuss the alternate formulation of the definition of two-sided limit, in terms of equality of the two one-sided limits. I think that should be discussed.)