e

Discovered by Euler 1737.

$\left(1+\frac{1}{x}\right)^x < e < \left(1+\frac{1}{x}\right)^{x+1}$

Remarkably, as shown in the graph below, as $x$ increases, $y=\left(1+\frac{1}{x}\right)^x$ increases, while $y=\left(1+\frac{1}{x}\right)^{x+1}$ decreases. The limit of both is $e = 2.71828182845904523536\dots$

Graph of $y=e^x$ and $y=1+x$ showing that $e^x>1+x$ when $x>0$ . The curves meet only when $x=0$ .

With very small $x>0$ , $e^x$ is extremely close to $1+x$ :
e x near 1

Illustration that $e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}\dots$

Graph showing the area under $y=\dfrac{1}{x}$ from $1$ to $e$ , which has an area of $1$ . i.e. $\displaystyle\int_1^e \dfrac{1}{x}\,\mathrm{d}x=1$
Areaunder1overx

$y=\displaystyle \sqrt[x]{x}$ reaches its maximum at $x=e$ .

Leave a Reply Cancel reply