Compound Interest Formula 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 $e$?
As 
$$
e = \lim_{n \to \infty} \left( 1 + \frac1n \right)^n,
$$
and as for any real number $r &gt; 0$, we have 
$$
\lim_{n \to \infty} \frac{n}{r} = \infty = \lim_{n \to \infty} n \  \mbox{ and } \ \lim_{n \to \infty} \frac{r}{n} = 0 = \lim_{n \to \infty} \frac1n,
$$
so we obtain
$$
\lim_{n \to \infty} \left( 1 + \frac{r}{n} \right)^n = \lim_{n \to \infty} \left( \left( 1 + \frac{r}{n} \right)^{\frac{n}{r}}  \right)^r = e^r.
$$

Hence we also conclude that for any real numbers $r &gt; 0$ and $t &gt; 0$, we have 
$$
\lim_{n \to \infty} \left( 1 + \frac{r}{n} \right)^{nt} = \lim_{n \to \infty} \left( \left( 1 + \frac{r}{n} \right)^{\frac{n}{r}}  \right)^{rt} = e^{rt}.
$$

Thus if $P_0$ is the initial amount, $r &gt; 0$ is the interest rate compounded $n$ times a given period of time, then after $t$ such periods of $t$, our amount will be
$$
P_0 \times \left( 1 + \frac{r}{n} \right)^{nt},
$$
and so if our interest at rate $r$ is compounded continuously, then our amount after $t$ number of periods will be
$$
P_0 e^{rt}.
$$

Hope this helps.
There is a formula for case of f(x)^g(x)
When f(x) tends to 1 then 
We can write e^((f(x)-1)*g(x))
It is applicable everywhere ..
It is a odd formula but very helpful