Question

• Show that \(\displaystyle\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n=e^x\).
• You may use the definition of \(\displaystyle e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n\).

Solution

Let \(m=\dfrac{n}{x}\)

$$ \lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n=\lim_{n\to\infty}\left(1+\frac{1}{m}\right)^{mx} $$

Notice as \(n\to\infty\), \(m\to\infty\).

Using the definition of \(e\).

$$ \begin{align*} \lim_{n\to\infty}\left(1+\frac{1}{m}\right)^{mx}&=\left(\lim_{m\to\infty}\left(1+\frac{1}{m}\right)^m\right)^x\\\\ &=e^x \end{align*} $$