One of the most important limits in calculus is

$$\bbox[#F2F2F2,5px,border:2px solid black] {\lim_{x\to0}\left(1+{x}\right)^{1/x}=e}$$

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 \(x\approx0\), the function \(y=(1+x)^{1/x}\) takes on values in the near neighborhood of \(2.718\cdots\), and therefore \(e\approx2.7182\).

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

\(x\) | \(y\) | \(x\) | \(y\) |
---|---|---|---|

\(100\) | \(1.04723\) | ||

\(10\) | \(1.27098\) | \(-0.99\) | \(104.762\) |

\(1\) | \(2.0000\) | \(-0.9\) | \(12.9155\) |

\(0.5\) | \(2.2500\) | \(-0.5\) | \(4.0000\) |

\(0.1\) | \(2.59374\) | \(-0.1\) | \(2.86797\) |

\(0.01\) | \(2.70481\) | \(-0.01\) | \(2.732\) |

\(0.001\) | \(2.71692\) | \(-0.001\) | \(2.71964\) |

\(0.0001\) | \(2.71815\) | \(-0.0001\) | \(2.71842\) |

Letâ€™s replace \(1/x\) by \(u\). So when \(x\to0\), we have \(u\to\infty\). Thus

\[\lim_{x\to0}(1+x)^{1/x}=\lim_{u\to\infty}\left(1+\frac{1}{u}\right)^{u}=e.\]

In other words:

$$\bbox[#F2F2F2,5px,border:2px solid black] {\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^{x}=e}$$