Auto-antonyms are words that can hold two, generally opposite, meanings at the same time. Once you know what they are, you’ll start to see them everywhere.
“Dust” is a good example. You can remove dust, like dusting a shelf, or you can add dust, like dusting a cake with powdered sugar.
It's possible for phrases to work this way too.
"Take care of" can mean to protect or look after someone or something, like a baby or a lawn. Consider this usage though: "An antibiotic should take care of that infection." Since you generally don’t want to protect or look after an infection, in this instance "take care of" means to get rid of or dispose of something.
So how did we get here? Let's start with the history of "care."
"Care" is Germanic. In Old English, it could refer to sorrow or mental anguish, as well as fear, doubt, or concern. You can still hear "care" used this way in phrases like "set your cares aside." In other words, don’t worry about anything.
In early meanings, "care" could also mean serious attention. We use that meaning when we say things like "I read the document with great care."
By the 16th century, "care" comes to refer to oversight with the purpose of protection or preservation. For example, you might leave your kids in someone else’s care while you’re at work. This is the version of "care" we see in "take care of."
The phrase "take care of" shows in up in the 1500s. In addition to look after, "take care of" also meant to deal with or dispose of something, like a problem. That may sound ominous, but taking care of something doesn’t have to end with someone sleeping with the fishes.
For example, your partner might offer to take care of the dishes tonight. That’s when you either pass out from shock or thank them and run out of the room before they change their mind.