I was trying to explain $e$ to someone and I ran across this idea. So, $e$ is typically defined as $$e =\lim_{n\to \infty} (1+1/n)^n $$ Which comes from the formula for compound interest $$P(t) = P_0\left(1+\frac{r}{n}\right)^{nt}.$$ I could not explain why it is that we divide by $n$. The typical answer is that you simply "divide" the interest rate across each compounding period, but of course means that the borrower will end up paying more. This just seems like trickery on the part of the lender. Does anybody have any idea why this formula in particular? Was it simply to trick lenders?
$e$ also then becomes kinda flukey. How would you justify that $$ \lim_{n\to \infty} \left( 1+\frac{x}{n}\right)^n$$ is an exponential function to some base let alone to the base $e$?