Well, I don't have my calculus textbook to hand, but I can tell you what I took from the class.

1. e is the exponential base for which f'(x) = f(x).

2. ln is the logarithm base e, and when f(x) = ln x, f'(x) = 1/x.

3. e is the sum of the series x^n / n! .

4. The textbook did specifically cover the fact that e is the limit of (1 + 1/n)^n as n goes to infinity, and it also specifically tied this in to the idea of computing interest by an obviously incorrect method. You could only call this a "natural mistake to make" in the same sense that it's "natural" to assume the square root of 10 must be 5, or that the geometric mean of two numbers is necessarily equal to the arithmetic mean.

5. However, the limit is important in that it illustrates that one to an infinite power is an indeterminate form.

6. As detailed in points (1) and (2), and hinted by the name "natural logarithm", we measure exponentials and logarithms by reference to e for the same reason we measure angles in radians.

It's possible that this particular definition of e is important to a proof of one of the properties of e^x or ln x, but if so I don't remember reading about it in the textbook and it wouldn't have been covered in class. In my real analysis class, we used the Maclaurin series for e; (1 + 1/n)^n was never mentioned.

(It's really easy to show that that series is monotonically increasing.)

> Then later when you have formally introduced sequences and how to prove convergence

This is not material you'd expect at all in a calculus class. If sequences are mentioned, it would only be in passing as you move to series. Several methods of testing infinite series for convergence are covered. What it means for a sequence to converge is not. Limits are not defined in terms of sequences. Infinite series would be covered after, not before, differential and integral calculus.

You have to have a name for e because otherwise it would be impossible to work with. But it is interesting and the wrong way to compute interest isn't; there's no point in trying to motivate something important with something unimportant.