4.12 The number e

One of the most important limits in calculus is

$$\underset{x\to 0}{lim}{(1+x)}^{1/x}=e$$

where $e$ is an irrational number (like $\pi$) and is approximately 2.718281828.

We are not going to prove that such a limit exists, but we will content ourselves by plotting $y=(1+x{)}^{1/x}$ (Figure 1), and show graphically that as $x\approx 0$, the function $y=(1+x{)}^{1/x}$ takes on values in the near neighborhood of $2.718\cdots$, and therefore $e\approx 2.7182$.

 

 

As $x$ approaches zero from the left, $y$ decreases and approaches $e$ as a limit. As $x$ approaches zero from the right, $y$ increases and also approaches $e$ as a limit.

As $x\to \mathrm{\infty }$, $y$ approaches the limit 1, and as $x\to -1$ from the right, $y$ increases indefinitely (see Table 1).

 

Let’s replace $1/x$ by $u$. So when $x\to 0$, we have $u\to \mathrm{\infty }$. Thus

$$\underset{x\to 0}{lim}(1+x{)}^{1/x}=\underset{u\to \mathrm{\infty }}{lim}{(1+\frac{1}{u})}^u=e.$$

In other words:

$$\underset{x\to \mathrm{\infty }}{lim}{(1+\frac{1}{x})}^x=e$$