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 \(e\approx2.7182\).

Figure 1. Graph of \(y=(1+x)^{1/x}\). Note that this function is not defined at \(x=0\).

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).

Tabel 1
\(x\) \(y\) \(x\) \(y\)
\(100\) \(1.04723\)
\(10\) \(1.27098\) \(-0.99\) \(104.762\)
\(1\) \(2.0000\) \(-0.9\) \(12.9155\)
\(0.5\) \(2.2500\) \(-0.5\) \(4.0000\)
\(0.1\) \(2.59374\) \(-0.1\) \(2.86797\)
\(0.01\) \(2.70481\) \(-0.01\) \(2.732\)
\(0.001\) \(2.71692\) \(-0.001\) \(2.71964\)
\(0.0001\) \(2.71815\) \(-0.0001\) \(2.71842\)

Let’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:

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

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