The number $e$

One of the most important limits in the Calculus is

$$\lim_{x \to 0} (1 + x)^{\frac{1}{x}} = 2.71828\cdots = e$$

To prove rigorously that such a limit $e$ exists, is beyond the scope of this book. For the present we shall content ourselves by plotting the locus of the equation

$$y = (1 + x)^{\frac{1}{x}}$$

and show graphically that, as $x \dot= 0$, the function $(1 + x)^\frac{1}{x}(= y)$ takes on values in the near neighborhood of $2.718\dots$, and therefore $e = 2.718\dots$, approximately.

Figure 3.14: The function $(1+x)^{1/x}$.

$x$	-.1	-.001	.001	.01	.1	1	5	10
$y=(1+x)^{1/x}$	2.8680	2.7195	2.7169	2.7048	2.5937	2.0000	1.4310	1.0096

As $x \to 0-$ from the left, $y$ decreases and approaches $e$ as a limit. As $x \to 0+$ from the right, $y$ increases and also approaches $e$ as a limit.

As $x \to \infty$, $y$ approaches the limit $1$; and as $x \to -1+$ from the right, $y$ increases without limit.

Natural logarithms are those which have the number $e$ for base. These logarithms play a very important rôle in mathematics. When the base is not indicated explicitly, the base $e$ is always understood in what follows in this book. Thus $\log_e\ v$ is written simply $\log\ v$ or $\ln\ v$.

Natural logarithms possess the following characteristic property: If $x \to 0$ in any way whatever,

$$\lim \frac{\log(1 + x)}{x} = \lim \log(1 + x)^{\frac{1}{x}} = \log\ e = \ln e % = 1.$$