Usage experts have been surprisingly passionate in their campaign against the rule banning all split infinitives. In her 1996 book Woe Is I, Patricia O’Conner gives the rule a tombstone icon and labels it RIP—rest in peace (182). Writing in 1975, William Morris and Mary Morris called the rule a “pedantic bogey” dating from the nineteenth century (338).
The New York Public Library Writer’s Guide to Style and Usage fills us in on the history:

Why have the experts fought so hard against the absolute ban on split infinitives? The reason is simple: They care about good writing. As R. H. Copperud puts it, “If a sentence doesn’t sound right, it isn’t any good, whether the sentence is split, rewoven, braided, or sawed in half” (202).
The effort to avoid the split infinitive at all costs has possibly resulted in more bad writing than the split infinitive itself. Consider the following examples, given by Rene J. Cappon.

Richard Lederer and Richard Dowis go so far as to hint that unsplitting an infinitive can at times be an error—because the revision changes the intended meaning or results in ambiguity. Consider their example of a correct split infinitive.

The following alternatives change the meaning because promptly attaches itself to the wrong word.

And this next alternative is ambiguous because we can’t tell whether promptly goes with decided or return.

There are times, of course, when a split infinitive sounds more awkward than alternative phrasing. Rene J. Cappon gives a few examples:

Unsplitting these infinitives results in more graceful phrasing: to increase profits substantially; to obey the law of the land at all times; to deal with the economic slump quickly and decisively.
William Morris and Mary Morris can have the last word: On the matter of split infinitives, they advise, “Proceed with caution, but not in fear” (338).