## University Calculus: Early Transcendentals (3rd Edition)

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 $x \lt 0$ $\lim\limits_{n \to \infty}(1+\frac{x}{n})^n=\lim\limits_{t \to -\infty}(1+\frac{1}{t})^{xt}=\lim\limits_{t \to -\infty}((1+\frac{1}{t})^{t})^x=e^x$
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 $x \lt 0$ $\lim\limits_{n \to \infty}(1+\frac{x}{n})^n=\lim\limits_{t \to -\infty}(1+\frac{1}{t})^{xt}=\lim\limits_{t \to -\infty}((1+\frac{1}{t})^{t})^x=e^x$