Many copy editors around the world, particularly in the USA, still appear to believe that the split infinitive is some kind of grammatical sin (despite what nearly all the quality handbooks of grammar and style actually say), so it may be useful to provide some additional commentary here.
By ‘split infinitive’ people mean the construction illustrated in to really succeed, where an adjunct (really) comes between the infinitival marker to and the plain form verb that is the head of the infinitival clause.
But the term ‘split infinitive’ is very misleading. English doesn't have an infinitive form of the verb in the way a language like French does. French succéder is a single word, but English to succeed is not; it's two words. The infinitival subordinating marker to is quite distinct from the verb. Verb phrases often begin with the head verb (as in the imperative Be careful!), but they can also begin with an adjunct such as an adverb (as in the imperative Really be careful!). No rule of English grammar requires that a verb phrase preceded by infinitival to must begin with its head verb — rather than with an adverb. And if you can begin a verb phrase with an adverb and mark that verb phrase with to, you automatically get phrases like to really succeed:
VP / \ / \ Sub VP | / \ | / \ | Adv VP to | | | | really succeed
All serious usage manuals and scholarly grammars recognise that phrases of this sort have occurred in English since its earliest history. They generally note that in many cases placing an adjunct between to and the verb is stylistically preferable to other orderings. However, people who have never investigated the matter tend to believe otherwise. It has never been fully clear why, but one source that may have been influential is an 1866 book called A Plea for the Queen's English by Henry Alford (a churchman who was at one time the Dean of Canterbury). Alford gives no evidence or argument at all, but simply cites a prejudice of his own that is contradicted by the testimony of another English speaker who wrote to him. He says (page 188):
238. A correspondent states as his own usage, and defends, the insertion of an adverb between the sign of the infinitive mood and the verb. He gives as an instance, "to scientifically illustrate." But surely this is a practice entirely unknown to English speakers and writers. It seems to me, that we ever regard the to of the infinitive as inseparable from its verb. And when we have already a choice between two forms of expression, "scientifically to illustrate," and "to illustrate scientifically," there seems no good reason for flying in the face of common usage.
That is the entirety of Alford's discussion. It simply seems to him that "surely this is a practice entirely unknown to English speakers and writers." His opinion was wrong then and still is.
To cite an arbitrary example from modern usage, there are over 650,000 occurrences of sentences on the web containing the sequence to at least try, as in The employer intended to at least try to explore possible reasonable accommodations for coping with disabled employees. They mostly change meaning or become ungrammatical if at least is shifted to a different position. Intended at least to... sounds as if it means at least intending, possibly more than intending; at least to explore sounds as if it means at least explore, possibly more than explore; if the intended meaning is about at least trying, and possibly more than just trying, the best order of words, without question, is the one that puts at least in between to and try, and that is the overwhelmingly most common order.
Occasionally it is impossible to avoid placing the adjunct in that position (see this Language Log post by Arnold Zwicky for a discussion of modifiers in infinitives that must separate the to from the verb; astonishingly, the American grammarian George O. Curme had already noted this as early as 1914).
There is no clear support for the people who insist that a ‘split infinitive’ is a grammatical error. Even quite conservative grammar and usage books agree that it is not, but citing evidence for the relevant sort of people may be useless: they may not be convinced by any amount of evidence. But for what it is worth, huge numbers of examples of split infinitives can be found in uncontroversially respectable and widely admired literature. Published examples have appeared in every subsequent century since the 14th. The list of writers who have been happy to use the construction is so extensive as to make it utterly absurd to call it ungrammatical: Robert Browning, Edmund Burke, Frances Hodgson Burnett, Lord Byron, Willa Cather, Marie Corelli, Daniel Defoe, John Donne, Sir Arthur Conan Doyle, Daphne du Maurier, George Eliot, William Faulkner, F. Scott Fitzgerald, Benjamin Franklin, John Galsworthy, Rider Haggard, Thomas Hardy, Ernest Hemingway, Herbert Hoover, Henry James, Jerome K. Jerome, Rudyard Kipling, Sinclair Lewis, Abraham Lincoln, Arthur Machen, Theodore Roosevelt, George Bernard Shaw, Sir Philip Sidney, Robert Southey, Herbert Spencer, Samuel Taylor Coleridge, James Thurber, Mark Twain, Oliver Wendell Holmes Sr., Woodrow Wilson, William Wordsworth... and hundreds of other novelists, statemen, poets, essayists, and literary critics (for a large selection of quotations and references, see George O. Curme, Syntax, D. C. Heath, 1930, §49 2 c, especially pages 461-465). We searched the text of two random books by novelists not listed above, one American and the other British, both from the late 19th century when many 20th-century prescriptive grammarians were growing up and becoming acquainted with literature. We underline the relevant verb and bracket its VP in the examples that we found:
1. From The Red Badge of Courage, by Steven Crane (1895), about the American civil war:
He tried to [mathematically prove to himself that he would not run from a battle].
He began to [blithely roar at his staff: "We'll wallop 'im now..."]
The youth had a thought that it would not be handsome for him to [freely condemn other men].
He waited as if he expected the enemy to [suddenly stop, apologize, and retire bowing].
2. From Dracula, a Victorian epistolary horror novel by Bram Stoker (1897):
I knocked gently and rang as quietly as possible, for I feared to disturb Lucy or her mother, and hoped to [only bring a servant to the door].
I said to him, "Go to Dr. Van Helsing, and ask him to [kindly come here at once].
And so we proceeded to [minutely examine them].
So well as I can remember, here it is: "I have studied, over and over again since they came into my hands, all the papers relating to this monster, and the more I have studied, the greater seems the necessity to [utterly stamp him out].
He seems to have power at these particular moments to [simply will], and her thoughts obey him.
(In that last example the verb will is not the modal verb but the ordinary lexical verb meaning something like "exercise volition".)
Obligatorily split infinitives.
14 May 2004)
Split decision. (Geoff Nunberg, 23 May 2004)
Two bites of authors' remorse. Geoff Pullum, 20 September 2004)
The pointless game of Grammar Gotcha. (Geoff Pullum, 11 April 2005)
Not to or to not. (Arnold Zwicky, 7 May 2005)
Obligatorily split infinitive in real life. (Geoff Pullum, 19 May 2005)
No splitting in court. (Arnold Zwicky, 14 July 2005)
Better to X than to not Y. (Eric Bakovic, 5 November 2005)
Joe, this is for you. (Geoff Pullum, 29 March 2007)
Irrational terror over adjunct placement at Harvard. (Geoff Pullum, 29 April 2008)
Books more loved than looked in. (Geoff Pullum, 30 April 2008)
Nonintervention. (Arnold Zwicky, 2 May 2008)
Contamination. (Arnold Zwicky, 10 May 2008)
Crazies win. (Arnold Zwicky, 13 May 2008)
Heaping of catmummies considered harmful. (Mark Liberman, 21 August 2008)
Minor writers, revolt!. (Geoff Pullum, 21 August 2008)
Inconsistent Latinophilia. (Geoff Pullum, 26 September 2008)
Economist still chicken: botches sentence rather than split infinitive. (Geoff Pullum, 11 June 2013)
At last: a split infinitive in The Economist. (Geoff Pullum, 17 June 2013)