One of the most common uses of infinitives is as the object of a preposition. When used in such a way, the infinitive is often the rough equivalent of the English gerund, that is, the "-ing" form of the verb, and can be translated in that way.
Using Infinitives After Prepositions in Spanish
In some cases, the infinitive is used to indicate additional action by the subject of the sentence:
Examples: Roberto salió sin verte. Roberto left without seeing you. Saldrá después de comer. She will leave after eating. Chile ganó por no seguir a la ortodoxia. Chile won by not following orthodoxy. Todos los niñitos se conformaban con aprender su letra de molde. All the children resigned themselves to learning how to print.
In other cases, the infinitive is used just as would any other noun in a prepositional phrase:
Examples: Gracias por no fumar. Thank you for not smoking. Para ellos, ser vasco es incompatible con ser español. For them, being Basque is incompatible with being Spanish. El presidente viajó a Londres para hablar de la situación humanitaria. The president traveled to London in order to speak about the humanitarian situation. Si tienes preguntas acerca de comprar la revista, favor de llamar por teléfono. If you have questions about buying the magazine, do the favor of calling by telephone.
If you're translating from English to Spanish, there are times you translate "-ing" verb forms with the -ando or -iendo verb forms of Spanish. For example, "I am speaking" can be translated as estoy hablando. However, when the verb follows a preposition, you should never translate using that form of the verb; use the infinitive instead.
English: I am sick of thinking about you.
Correct: Estoy harta de pensar en ti.
Incorrect: Estoy harta de pensando en ti.
There is one common usage of the infinitive following a preposition in Spanish that doesn't have an exact English equivalent. The infinitive acts as something of a passive description:
Examples: La lata sin abrir puede durar hasta 12 meses. An unopened can can last up to 12 months. En la mesa estaba una manzana a medio comer. On the table was a half-eaten apple. Hay muchas tareas por hacer. There are many tasks to be done. Tengo un par de libros sin leer. I have a pair of unread books.
Such a use of the infinitive is most common following sin (meaning "without") and the phrase a medio (translated as "half").