Here's how Fowler introduced this subject in the 1920s, in Modern English Usage:
The English-speaking world may be divided into (1) those who neither know nor care what a split infinitive is; (2) those who do not know, but care very much; (3) those who know & condemn; (4) those who know & approve; & (5) those who know & distinguish.

1. Those who neither know nor care are the vast majority, & are happy folk, to be envied by most of the minority classes; 'to really understand' comes readier to their lips & pens than 'really to understand', they see no reason why they should not say it (small blame to them, seeing that reasons are not their critics' strong point), & they do say it, to the discomfort of some among us, but not to their own.

In the second class are the people Fowler most strongly condemns: those who have heard there is a bad thing called a split infinitive, but don't know what it is. "These people betray by their practice that their aversion to split infinitive springs not from instinctive good taste, but from tame acceptance of the misinterpreted opinion of others."

By this he means people who would write "really to be understood" or "to be understood really", both of which are ugly, in order to escape the imagined fault of splitting the verb complex and writing "to really be understood" or "to be really understood". The last two ways are the good, plain English ways of saying it, yet people in class 2 imagine they are taboo. Yet only the first of the two is a "split" infinitive: there isn't even an imaginary rule against the second.

Those in class 3, who know what it is and don't want to do it, can be recognized by the weird distortions they come up with. A word is in the wrong place, a phrase reads wrong, it is unnatural, there is something stylistically awkward about it: why would anyone write this? Aha! They're trying to avoid a split infinitive. So they twist away from natural English because they're following this imaginary "rule".

Example: "Every effort must be made to increase adequately professional knowledge & attainments." Huh? Adequate professional knowledge...? Oh, no, I see, what they meant was "to adequately increase professional knowledge". If you say it in the natural English way, it's so much clearer. Avoiding the split is unnatural and ugly and can read like nonsense.

The problem with the fourth class, who deliberately use the split infinitive, is that so many people (including, unfortunately, editors and proofreaders and teachers) are aware that it might be frowned on that they are prepared to advise "better not do it", just in case someone thinks you're ill-educated.

In my opinion it would be far better if teachers and editors actively encouraged plain English, and struck out queer circumlocutions like "to increase adequately": if they insisted that splitting infinitives is the good, correct English way. Note: not merely tolerated, not merely letting you off a rule which is supposed to exist, but coming out and saying very firmly there is no such rule.

It is a superstition. It is a fetish. If you're a native-born speaker of English, then you know all the true rules of the language instinctively. If someone comes up to you and springs a new rule on you, one you have to mind as an adult, they're wrong. There is no such rule.

You don't find these fictional rules in grammar-books. You don't find them in dictionaries. Grammars and dictionaries are there for describing what people actually say: it's been a hundred years since a respectable work of authority laid down the law (or rather claimed to).

What grammarians and lexicographers say hasn't filtered down to teachers yet, and you still get teachers saying that such-and-such is a rule, or is bad. Well the teacher is wrong.