I’m going to quickly give you an example.
That’s it! I just gave it. To give is the infinitive form of the verb give, and I have split it with quickly.
Spitting an infinitive is not a crime against grammar, but it’s not the most elegant construction either. We do it all the time in spoken word – I’m just going to quickly brush my teeth and I’m ready – but in writing it can be a bit clunky. A better option for that opening sentence would be: I’m going to give you a quick example.
The ‘rule’ of not splitting the infinitive is a bit fuddy duddy, but it’s a useful thing to understand.
For an amusing explanation of the history of this ‘rule’, check out the fabulous Grammar Girl.