# What is the limit of (1 + 2/x)^x as x approaches infinity?

May 2, 2016

${\lim}_{x \rightarrow \infty} {\left(1 + \frac{2}{x}\right)}^{x} = {e}^{2}$

#### Explanation:

Use ${\lim}_{u \rightarrow \infty} {\left(1 + \frac{1}{u}\right)}^{u} = e$

${\lim}_{x \rightarrow \infty} {\left(1 + \frac{2}{x}\right)}^{x} = {\lim}_{x \rightarrow \infty} {\left(1 + \frac{1}{\frac{x}{2}}\right)}^{x}$

$= {\lim}_{x \rightarrow \infty} {\left({\left(1 + \frac{1}{\frac{x}{2}}\right)}^{\frac{x}{2}}\right)}^{2}$

$= {\left({\lim}_{x \rightarrow \infty} {\left(1 + \frac{1}{\frac{x}{2}}\right)}^{\frac{x}{2}}\right)}^{2}$

Now we have the form above with $u = \frac{x}{2}$, so we evaluate the limit.

$= {e}^{2}$

May 4, 2016

Jim, awesome ..

What this limit really represents is essentially the horizontal asymptote y = ${e}^{2}$, reflecting the function's long term graphical behavior.

#### Explanation:

Here are a couple of TI screenshots showing the graph and the decimal expansion for ${e}^{2}$.

If we went even further out to the right and then asked some random guy on the street if they are looking at a straight line, they would say "yes!".

But in fact you are looking at the curve endowed with concavity, not a straight line. The curve is asymptotically approaching the value of $y = {e}^{2}$