Why is continuous compounding not a great deal better than say, monthly or annual compounding?

Question

I thought that if you are adding interest to an amount continuously (every moment), it would definitely gain more than having the same amount being compounded only monthly or annually.
the equation for continuous compounding is S = Pe^(j-infinity*t)
S = future value
P = principal amount
e = natural log
j-infinity = force of interest (also denoted with delta)
t = time

Answer 1

Let's say you have an account making 5% per year, compounded continuously. This is equivalent to (e^(.05) - 1)% compounded annually, or approximately 5.1271%. This is not very much larger than 5%, and if you are given a choice between, say, 5% compounded continuously and 5.2% compounded anually, you should absolutely choose the 5.2%. Here, the small increase in interest rate is far more important than continuous compounding. In order for continuous compounding to offer you even 1 extra percent over annual compounding, you would have to have an interest rate in excess of 13.81651%

Note however, that if you happen to have an extremely high interest rate (say you're a credit card company and get to charge your customers an exhorbitant 25% interest), then continuous compounding offers a much larger advantage over annual compounding (in this case, it would be equivalent to 28.40254% compounded annually). In an extreme (and hopefully nonexistent) case of 100% interest, continuous compounding offers the equivalent of 271.82818% ! So the advantage depends on what your interest rate is in the first place.

Answer 2

because the limit of this compounding is not very big,
it is e.
and by the way, e is not the natural log, it is the exponential function (which is exactly the inverse of the natural log)