Are the following both okay? Is there any difference?
A life skill is one that is necessary or extremely useful to manage well in daily life.
A life skill is one that is necessary or extremely useful for managing well in daily life.
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That's more tricky than you might think. At first glance to manage is an infinitive form of the verb, and it's saying what is useful. As such, it would be an example of the common form adjective + infinitive verb. Here the verb gives content to the adjective. For example:
I was happy to go on holiday.
We were sad to see our team lose the football match.
But look closer and you will see that's not the case here. Compare these two sentences:
A life skill is useful to have.
A life skill is useful to manage.
Here we would say sentence 1 is complete in itself, but sentence 2 doesn't make sense. You are left thinking, "manage what?" We don't manage a life skill.
What's really happening is that in this case we have a purpose clause. To manage really means "in order to manage".
So the first sentence is really saying:
A life skill is one that is necessary or extremely useful in order to manage well in daily life. (or, "so that we can manage well...")
As written, it's not wrong, because "in order to" can be shortened to "to". But because it has the potential to be confusing, I would use the second sentence. "Useful for managing" clearly implies purpose; it can only mean one thing.