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

e graph
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.
e^x and 1+x

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

y=\displaystyle \sqrt[x]{x} reaches its maximum at x=e.
e as xp1ox

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