Let's examine some integer values of \(x\) to see the limiting value of this expression.
| \(x\) |
\(1\) |
\(2\) |
\(3\) |
\(\ldots\) |
|
\[\begin{align*} y&=\left ( 1+ \dfrac{1}{x} \right )^x \\ \end{align*}\]
|
\[\begin{align*} \left (1+\dfrac{1}{1} \right )^1 &= \left ( \dfrac{2}{1}\right )^1 \\ &=2 \end{align*}\]
|
\[\begin{align*}\left (1 + \dfrac{1}{2} \right )^2 &=\left ( \dfrac{3}{2} \right)^2 \\ &=2.25 \end{align*}\]
|
\[\begin{align*} \left (1 +\dfrac{1}{3} \right )^3 &= \left ( \dfrac{4}{3} \right )^3 \\ &\approx 2.3704 \end{align*}\]
|
\[\begin{align*} &\ldots \\ &\phantom{\ldots} \\ &\ldots \end{align*}\]
|
| \(\ldots\) |
\(100\) |
\(1000\) |
\(10 000\) |
|
\[\begin{align*} &\ldots \\ &\phantom{\ldots} \\ &\ldots \end{align*}\]
|
\[\begin{align*}\left (1+\dfrac{1}{100}\right )^{100} &= \left ( \dfrac{101}{100}\right )^{100} \\ &\approx 2.7048 \end{align*}\]
|
\[\begin{align*} \left (1+\dfrac{1}{1000} \right )^{1000} &= \left ( \dfrac{1001}{1000}\right )^{1000} \\ &\approx 2.7169 \end{align*}\]
|
\[\begin{align*}\left ( 1+\dfrac{1}{10\,000}\right )^{10\,000} &= \left ( \dfrac{10\,001}{10\,000} \right )^{10\,000} \\ &\approx 2.7181 \end{align*}\]
|
We see that as \(x\) becomes larger, the value of the expression changes by a smaller and smaller amount.
In fact, the change can be shown to approach \(0\).
In other words, the expression is approaching a limiting value.
The limiting value of this expression is the irrational number \(e=2.718281828459\ldots\), a non-terminating decimal.
By making the substitution \(t=\dfrac{1}{x}\), we get the following alternate definition of Euler's number:
\[e = \lim_{x\to \infty}\left(1+\dfrac{1}{x}\right)^x=\lim_{t\to 0} (1+t)^{\frac{1}{t}}\]