1. to leave (a matter, decision, etc. to someone); to entrust (to someone); to entrust (someone) with
ここは彼女に任せよう。Let's leave this to her.
2. to leave (to chance, one's imagination, etc.); to let (take its own course); to let (happen); to give (oneself to something)
あとは自然に任せればいい。You can just leave the rest to nature.
3. to use (money, strength, spare time, etc.) to the full; to use freely; to use without reserve
as ...に任せて
力に任せてドアを押した。I pushed the door with all my strength.