jmc

Recall the identity $\mathrm{lcm}[a,b]\cdot \gcd (a,b)=ab$, which holds for all positive integers $a$ and $b$. Thus, $$\mathrm{lcm}[9999,100001]=\frac{9999\cdot 100001}{\gcd (9999,100001)},$$so we focus on computing $\gcd (9999,100001)$.

Notice that $100001=99990+11=10(9999)+11$. Therefore, any common divisor of $100001$ and $9999$ must be a divisor of $100001-10\cdot 9999=11$. The possibilities are $1$ and $11$.

In fact, $9999=11\cdot 909$, so $11$ is a divisor of $9999$ and $100001$, which gives $\gcd (9999,100001)=11$.

Therefore, $$\begin{array}{rl}\mathrm{lcm}[9999,100001]& =\frac{9999\cdot 100001}{11}\\ & =909\cdot 100001\\ & =909\cdot 100000+909\\ & =\boxed{90,900,909}.\end{array}$$