1. To travel the world is my lifelong dream. (The infinitive phrase acts as the subject of the sentence)
2. She studied hard to pass the exam. (The infinitive phrase acts as the purpose of the verb "studied")
3. He went to the store to buy some milk. (The infinitive phrase acts as the purpose of the verb "went")
4. To be honest, I don't like the new design. (The infinitive phrase acts as an adverb modifying the sentence)
5. The best thing to do is to wait patiently. (The infinitive phrase acts as an adjective modifying the noun "thing")