Answer:
Sure, here are the infinitives, gerunds, and participles identified in the sentences:
Sentence 1: "I don’t feel like studying English today."
- Infinitive: "to study"
- Gerund: "studying" (used as the object of "like")
Sentence 2: "We were careful to avoid the broken glass."
- Infinitive: "to avoid"
- Gerund: None
- Participles: "broken" (used as an adjective to modify "glass")