Final answer:
The men spread lime around Miss Emily's house to eliminate the odor of decay. It was a practical action to address a community problem, not to investigate, cover up a crime, or sabotage Emily's reputation.
Step-by-step explanation:
The passage describes men sprinkling lime around Miss Emily's house late at night. The presence of Miss Emily as a motionless figure, observing this action from her window, along with the fact that the smell eventually went away, alludes to the idea that the lime was used to neutralize an unpleasant odor. Considering the context provided by the rest of the story not included here, the most fitting answer to the question would be:
The men spread lime around Emily's house in order to get rid of the odor of decay.
This action is not principally about looking for a missing person or covering up a crime directly, nor is it meant to sabotage Emily's reputation. Rather, it is a practical measure to address a problem concerning the community, which is hinted at by the smell that eventually dissipated.