Answer
$\lim\limits_{n \to \infty}(1+\frac{x}{n})^{n}=e^{x}$
Work Step by Step
Given: $\lim\limits_{n \to \infty}(1+\frac{x}{n})^{n}=e^{x};x>0$
From the definition of derivative as a limit, we get
$f'(y) =\lim\limits_{h \to 0}\frac{ln(y+h)-lny}{h}$
Apply logarithmic property.
$\lim\limits_{h \to 0}\frac{\frac{ln(y+h)}{y}}{h}=\frac{1}{y}$
Suppose $h=\frac{1}{n}$ and $y=\frac{1}{x}$
when ${h \to 0}$ then ${n \to \infty}$
Thus,
$\lim\limits_{h \to 0}\frac{\frac{ln(1/x+1/n)}{1/x}}{1/n}=\frac{1}{1/x}$
$\lim\limits_{n \to \infty}(1+\frac{x}{n})^{n}=x$
Take exponent on both sides of above expression.
$e^{\lim\limits_{n \to \infty}ln(1+\frac{x}{n})^{n}}=e^{x}$
Since, $e^{lnx}=x$
Hence,
$\lim\limits_{n \to \infty}(1+\frac{x}{n})^{n}=e^{x}$
