The tortoise ran slowly and steadily to win the race.

to win the race -----is the infinitive phrase

I am thinking tortoise ran for what I am saying the phrase is used as noun

"to win the race" is the infinitive phrase, yes.

It's not being used as a noun, though.
Read the explanations and examples here, and re-think your answer.
http://grammar.ccc.commnet.edu/grammar/phrases.htm#infinitive

Actually, the phrase "to win the race" functions as an adverbial infinitive phrase modifying the verb "ran." It describes the purpose or goal of the tortoise's action.

To identify the role of a phrase in a sentence, you can follow these steps:

1. Identify the phrase: In this case, the phrase is "to win the race."

2. Determine the type of phrase: The phrase starts with the infinitive marker "to," indicating that it is an infinitive phrase.

3. Analyze its function: Considering the context, you can see that the phrase is not acting as a noun but rather modifying the verb "ran." It explains why the tortoise was running.

By breaking down the sentence and analyzing the structure and function of the phrase, you can determine how it contributes to the overall meaning of the sentence.