Step 1: Recall the common nickname. New York City has carried the nickname "Big Apple" since the 1920s, when a sports writer used it to describe the city as the top prize in racing circles.
Step 2: Check the origin. The name later got a boost from a 1970s tourism campaign that used "Big Apple" to promote the city, and it stuck in everyday use worldwide.
Step 3: Rule out the other choices. "Big Sweet Lime" and "Big Grape" are not real nicknames for any major US city, so they can be dropped right away. "None of the above" fails since a genuine nickname does exist.
Final Answer: New York City is known worldwide as the Big Apple. \[ \boxed{\text{Big Apple}} \]