Answer:
at any cost
Step-by-step explanation:
In order to identify "prepositional phrases," you have to know what "prepositions" are at first. Prepositions are meant to show relationships of a word to another in a sentence. Examples of prepositions are: at, about, in, to, along, off, near, above, except, up, but, against, during, until, etc.
You can immediately identify a prepositional phrase because it includes a preposition. In the example sentence above, the preposition is "at." Its object is "any cost." Therefore, the prepositional phrase is "at any cost." It functions as an adverb phrase modifying the degree of finishing the race.