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.
Keeping infinitives intact is actually a sensible idea. Otherwise you run the risk of writing sentences that sound like this:
- We wanted to, because we felt it was important, talk to you today about our water ski catalog.
Still, no grammarian today sees any value in having an official sanction against splitting infinitives, and everyone agrees that it was a silly rule to adopt in the first place. Even if the rule didn't exist, split infinitives would rarely occur; that's because we rarely split them in conversation.
On the other hand, there are certain situations in which splitting the infinitive produces precisely the effect you want to produce, which is to put less emphasis on the action conveyed in the infinitive and more on the modifier.
- I would now like you to slowly and precisely tell me what happened and how it happened. (Splitting the infinitive positions the adverbs slowly and precisely immediately before the verb tell and puts the emphasis on these two words.)
Source: Grammar for Smart People, by Barry Tarshis.