jmc

First, we find the repeating decimal expansion of 5/14: $$\frac{5}{14}=\frac{5}{5}\cdot \frac{5}{14}=\frac{25}{70}=\frac{25}{7}\cdot \frac{1}{10}=(3.\overline{571428})(0.1)=0.3\overline{571428}.$$The $1314^{\text{th}}$ digit after the decimal point is the $1313^{\text{th}}$ digit in the 6-digit repeating block 5-7-1-4-2-8. Since $1313÷6$ leaves a remainder of 5, our answer is the $5^{\text{th}}$ digit in the 6-digit block, which is $\boxed{2}$.