smc

The list starts with $12345$. There are $\left(\genfrac{}{}{0ex}{}{8}{4}\right)=70$ four-digit rising numbers that do not begin with $1$, and thus also $70$ five digit rising numbers that do begin with $1$ that are formed by simply putting a $1$ before the four digit number. Thus, the $71^{\text{st}}$ number is $23456$. There are $\left(\genfrac{}{}{0ex}{}{6}{3}\right)=20$ three-digit rising numbers that do not begin with a $1,2$ or $3$, and thus $20$ five digit rising numbers that begin with a $23$. Thus, the $91^{\text{st}}$ number is $24567$. Counting up, $24568,24569,24578,24579,24589,24678$ is the $97^{\text{th}}$ number, which does not contain the digit $\boxed{5}$.