The well-known grammarians' rule against splitting an infinitive means that nothing must come between to and the verb. It is a cramping rule; it increases the difficulty of writing clearly; it makes for ambiguity; and I know of no justification for its existence that can be put in the scale against these disadvantages. The commonest way in which it creates ambiguity is this: that if, let us say, you are forbidden to write "he failed to completely convince them" you will probably write "he failed completely to convince them". Your reader will then not be sure whether you mean "he completely failed to convince" or "he failed to completely convince", and they are obviously very different things.
Nor is this all. The split-infinitive taboo, leading as it does to the putting of adverbs in awkward places, is so potent that it produces an impulse to put them there even though there is not really any question of avoiding a split infinitive. I have myself been taken to task by a correspondent for splitting an infinitive because I wrote "I gratefully record". He was, no doubt, under the influence of the taboo to an exceptional extent. But sufferers from the same malady in a milder form can be found on every hand. We cannot doubt that the writer of the sentence "they appeared completely to have adjusted themselves to it" put the adverb in that uncomfortable position because he thought that to write "to have completely adjusted" would be to split an infinitive. The same fear, probably subconscious, may also be presumed to account for the unnatural placing of the adverb in "so tangled is the web that I cannot pretend for a moment that we have succeeded entirely in unweaving it". In this there is no possibility of splitting an infinitive because there is no infinitive. But the split infinitive bogy is having such a devastating effect that people are beginning to feel that it must be wrong to put an adverb between any auxiliary and any part of a verb, or any preposition and any part of a verb.
The infinitive can be split only by inserting a word or words between the word which, with to, forms the infinitive of the verb. "To fully understand" is a split infinitive. So is "to fully have understood". But "to have fully understood" is not.
In Plain Words I wrote of the rule against the split infinitive:
Still, there is no doubt that the rule at present holds sway, and on my principle the official has no choice but to conform; for his readers will almost certainly attribute departures from it to ignorance of it, and so, being moved to disdain of the writer, will not be "affected precisely as he wishes".
A friend whose opinion I value has reproached me for this, making no secret of his view that I am little better than a coward. I ought, he tells me, to have the courage of my convictions. I ought to say about the split infinitive, as I said about the "inanimate whose", that it is right for the official to give a lead in freeing writers from this fetish. The furthest I ought to allow myself to go along the road of safety-first is, according to him, to say that it is judicious for an official to avoid splitting whenever he can do so without sacrificing clarity, ease, and naturalness of expression. But rather than make that sacrifice he should resolutely split.
My friend may be right.