You split a banana with ice cream, fudge sauce, and whipped cream. You split an infinitive by inserting a modifier--an adverb, usually--between the to and the verb, as in "I want you to carefully read over these instructions." The notion that this incision is grammatically unsound was first set forth in the mid-1800s, and it finds its basis in Latin, a language in which the infinitive is a one-word verb form.