Purpose (In Order To / So That)

The aim is something you cannot simply decide to do: hearing better, not forgetting, a door not closing.

Verb ように vs Verb ために: Use ように. It pairs with potential forms, negatives and verbs describing states, because the goal is a condition you hope holds. ために needs a deliberate act with the same subject, so 聞こえるために is wrong where 聞こえるように is right.

Writing about a purpose pursued with unusual determination.

～んがために vs Verbる べく: んがために is the more emphatic and more archaic of the two, and it colours the purpose as single-minded, sometimes to a fault. べく is the neutral literary 'in order to' and is far commoner in ordinary formal prose.