Answer
Given function: $f(x)=(1+\frac{x}{n})^n$
Put $n=xt$ and take infinite limit:
Condition 1 when $x \gt 0$
$\lim\limits_{n \to \infty}(1+\frac{x}{n})^n=\lim\limits_{t \to \infty}(1+\frac{1}{t})^{xt}=\lim\limits_{t \to \infty}((1+\frac{1}{t})^{t})^x=e^x$
as$ \lim\limits_{t \to \frac{+}{-}\infty}((1+\frac{1}{t})^{t})=e^x$
Condition 2 when $x \lt 0$
$\lim\limits_{n \to \infty}(1+\frac{x}{n})^n=\lim\limits_{t \to -\infty}(1+\frac{1}{t})^{xt}=\lim\limits_{t \to -\infty}((1+\frac{1}{t})^{t})^x=e^x$
Work Step by Step
Given function: $f(x)=(1+\frac{x}{n})^n$
Put $n=xt$ and take infinite limit:
Condition 1 when $x \gt 0$
$\lim\limits_{n \to \infty}(1+\frac{x}{n})^n=\lim\limits_{t \to \infty}(1+\frac{1}{t})^{xt}=\lim\limits_{t \to \infty}((1+\frac{1}{t})^{t})^x=e^x$
as$ \lim\limits_{t \to \frac{+}{-}\infty}((1+\frac{1}{t})^{t})=e^x$
Condition 2 when $x \lt 0$
$\lim\limits_{n \to \infty}(1+\frac{x}{n})^n=\lim\limits_{t \to -\infty}(1+\frac{1}{t})^{xt}=\lim\limits_{t \to -\infty}((1+\frac{1}{t})^{t})^x=e^x$
