Complement (v. and n.) – to enhance or add extra features to make something better.
That tomato sauces complements my dinner.
Your new purple hat is the perfect complement to your ball gown.
Complement (n.) – the number needed to complete a group.
We have a full complement at the moment. Couldn't jam another sailor in if you tried.
Compliment (v.) – say something nice about someone. And this is the word to use if you’re thinking of ‘compliment slips’.
May I compliment you on your choice of purple hat and ball gown?
Complimentary (adj.) – free of charge.
Here’s your complimentary moist towelette.
Get it right
There's a ‘lime’ in the middle of the ‘saying something nice’ and ‘free gift’ compliment, so perhaps you can fix in your head the idea of praising a small green citrus fruit which someone has given you for free.