Step 1: Understanding the Question:
We need to choose the correct preposition of time to complete the sentence. The sentence is in the present perfect continuous tense ("has been living").
Step 2: Detailed Explanation:
The present perfect continuous tense is used to describe an action that started in the past and is still continuing in the present. This tense is often used with 'for' and 'since'.
For is used to specify a duration or period of time (e.g., for five years, for two months, for a long time).
Since is used to specify the starting point of the action (e.g., since 2010, since last Monday, since 8 AM).
In the given sentence, "2010" is a specific point in time when the action of living in the city began.
Therefore, the correct preposition to use is "since".
The complete sentence is: "He has been living in this city since 2010."
Step 3: Final Answer:
The appropriate preposition is "since".