I was reading a calculus book, and at some point the following standard limit was mentioned:
$$\lim_{n\to\infty}\left(1+\frac1n\right)^n=e$$
Afterwards, the author of this book invited the reader to try and compute, using a calculator, the value of the expression
$$\left(1+\frac1n\right)^n$$
for various inputs. He then stated that the outputs would look something like this:
$$\begin{array}{c|c}
n&(1+1/n)^n\\
\hline
10^3&2.7169239\\
10^4&2.7181459\\
\dots&\dots\\
10^{11}&1\\
10^{12}&1
\end{array}$$
The author then invites the reader to try and work out why this happens.
I thought this was an interesting question and one that actually shows how calculus can make concepts work that wouldn't normally be quantifiable or measurable in a physical world.
My theory is that, since any calculator has a certain precision that it can store values up to, after a certain point the expression
$1/n$ just gets rounded down to $0$, aren't there being enough bits to store the floating point value.
The whole expression is then evaluated as $1^n$ which is constantly equal to $1$. It doesn't matter if eventually the calculator can't keep up with the value of the exponent either and rounds it too up to something else, because the value of the power will be $1$ regardless of what the exponent ends up being.
Any other theories?
Best Answer
Your theory is the only one that actually happens in the real world, where all machines have finite precision and cannot represent numbers sufficiently close to $1$, like $1+1/n$. There is no other plausible explanation for why the computed expression suddenly drops to $1$.