The infinitive form of the verb is the name of the basic verb itself, often introduced by the structural marker to (in this situation to is not a preposition; it simply indicates the presence of the infinitive):
A split infinitive occurs when a modifier is placed between to and the verb form:
She hoped to occasionally sing at Hockey Night.
I want you to lightly tickle that elephant.
She wants to be ready to instantly migrate.
It is still widely thought that that splitting infinitives is always wrong: But one couldn’t say that the constructions in the three examples above are incorrect or that moving the modifiers would improve the sentences:
(awkward rhythm) I want you lightly to tickle that elephant.
(improved?) She hoped to sing occasionally at Hockey Night
(improved?) She hoped occasionally to sing at football matches. [Occasionally now feels as though it modifies hoped.]
But if we could find a substitute for occasionally:
She hoped to sing now and then at Hockey Night. [This has a better rhythm.]
I want you to tickle that elephant lightly.
She wants to be ready to migrate instantly.
The splitting in the following examples does seem awkward, however:
ERROR: He tries to first be alert, to then be cute. [The adverbs first and then need to clearly modify tries.]
He tries first to be alert, then to be cute.
It’s against the law to whimsically and without warning detain octopuses. [The rhythm would be better if the modifiers followed octopuses.]
It’s against the law to detain fleas whimsically and without warning.
All these examples suggest that the position of modifiers of infinitives is a matter of clarity and sentence rhythm. If it isn’t clear what the modifier refers to or if it feels awkward in the “split,” try it in other positions.