The sentences that include infinitives are:
- We always believed we had the ability to become doctors.
- To buy the expensive vase seemed foolish when they still had debt.
- They all wanted to fly across the country.
The other sentences do not contain infinitives. Specifically, "She presented the gift to him last night." does not include an infinitive, and "Arriving late is unacceptable to our principal." includes a gerund, not an infinitive.