(i) The functional equation of exp:

(ii) $\exp(-x)=1/\exp(x)$ for all $x\in{\mathbb{R}}.$

(iii) $\exp(p/q)=e^{p/q}$ for any integers $p$ and $q$ with $q\neq 0$.

(iv) $\exp x>0$ for all real $x$.

(v) $\exp x\rightarrow\infty$ as $x\rightarrow\infty$; whereas $\exp u\rightarrow 0$ as $u\rightarrow-\infty$.

Proofs (i) One needs to multiply the series together.

(ii) $\exp 0=1+0+\dots=1;$ by (i), we also have $\exp(x)\exp(-x)=\exp 0=1$, so $\exp(x)\neq 0$ and $\exp(-x)=1/\exp(x).$