jmc

The decimal representation of $\frac{1}{13}$ is $0.\overline{076923}$. Since the first six digits repeat, we know that after every 6th digit the pattern will restart. Since $43÷6=7r1$, the first 42 digits will be seven repetitions of the same pattern followed by the first digit of the pattern. Since the first digit is $\boxed{0}$, this is our final answer.