Step 1: Understanding the Concept:
The correct preposition to indicate a specific time frame or deadline is 'for' (for indicating a time period) and 'in' (for a specific month or time reference).
Step 2: Detailed Explanation:
The phrase 'the closing date for applications' correctly uses 'for' to indicate the period or deadline for the applications, and 'in this month' uses 'in' to denote the time frame within the current month.
Step 3: Final Answer:
Therefore, the correct prepositions are "for" and "in".