Another Way to Define e

Math Calculus

Limit to solve:

$$\underset{x\to 0}{\lim }\frac{e^x-1}{x}$$

Let $t=e^x-1$

$$\underset{t\to 0}{\lim }\frac{t}{\ln (t+1)}$$ $$\underset{t\to 0}{\lim }\frac{1}{\frac{1}{t}\ln (1+t)}$$

Inverse power log rule

$$\underset{t\to 0}{\lim }\frac{1}{\ln (1+t{)}^{\frac{1}{t}}}$$

Definition of e

$$\frac{1}{\ln e}$$ $$1$$