jmc

By AM-HM, $$\frac{a+b}{2}\ge \frac{2}{\frac{1}{a}+\frac{1}{b}}.$$Hence, $$\frac{1}{a}+\frac{1}{b}\ge \frac{4}{a+b}=4.$$Equality occurs when $a=b=\frac{1}{2}.$

Note that as $a$ approaches 0 and $b$ approaches 1, $\frac{1}{a}+\frac{1}{b}$ becomes arbitrarily large. Therefore, the set of all possible values of $\frac{1}{a}+\frac{1}{b}$ is $\boxed{[4,\infty )}.$