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Check out this dialogue between two calculus students (based on a true story):

Devyn
Riley, I’ve been thinking about the interest on my bank account.
Riley
So, like the compound interest formula,
Devyn
Yes! To make it easier to work with, suppose I deposit just $1, so $P=1$. Riley A$1 bank account balance? I can imagine that!
Devyn
If the interest is compounded monthly, $n=12$, and if the deposit is at $3\%$ interest, then $r = 0.03$.
Riley
Since $\dfrac {0.03}{12} = .0025$, that brings you to
Devyn
My question is: How long does it take the account to double in value? How long until the account balance is \$2?
Riley
Can’t we just setup
Devyn
Of course, but how to solve that for $t$? The variable is up in an exponent.
Riley
Hmmmm. I’m not sure…
What kind of operation will allow us to solve for $t$?
Take the square root of both sides. Take the cosecent of both sides. Raise both sides to the $1.0025$-th power. Take a logarithm of both sides.
Take the compound interest formula above $\displaystyle A = P\left ( 1 + \frac {r}{n}\right )^{nt}$ with $P = 1$, $r = 1$, and $t = 1$: Plug in larger and larger values of $n$ and see what happens to the values of $A$.
$\dfrac {1}{n}$ tends to $0$, so $\displaystyle \lim _{n \to \infty } \left (1 + \frac {1}{n} \right )^n = 1$. The exponent is getting bigger and bigger, so $\displaystyle \lim _{n \to \infty } \left (1 + \frac {1}{n} \right )^n = \infty$. $\displaystyle \lim _{n\to \infty } \left ( 1 + \frac {1}{n} \right )^n = 3$. $\displaystyle \lim _{n\to \infty } \left ( 1 + \frac {1}{n} \right )^n = 2.71828\ldots$.