Absence and nothingness with 無
無 means a state where an object or action lacks purpose, value, or effect. It expresses that no matter how hard you try, it will not lead to results, or that something makes no sense (nonsense).
Ability and Knowledge
無能 (Incompetent)
無能 means incompetent or incapable. It can describe a person or the lack of ability itself.
It combines 無 (nothing) with 能 (ability) to show lack of ability.

Police detective Yokomizo isn’t about to let Sasai, his prime murder suspect, leave. Sasai reminds the officer that he has photographic proof that he wasn’t near the victim at the time of the murder. He calls for Detective Mouri to back him up.
- 笹井
( : - 「名探偵
( のあなたからもいってやってくださいよ。 この刑事( がいかに無能( かという事( を…」- “You, the great detective, should tell him yourself. Just how incompetent this police detective is...”
Limits and Possibility
無理 (Impossible)
無理
It combines 無 (without) with 理 (reason) to show something without reasoning.

On a summer day, Hori and her friends in class can’t ignore the heat.
- 由紀
( : - 「外
( の暑( さってなんなの…? クーラーないと死( んじゃうって…っ」- “What's with this heat outside...? I'd totally die without the AC...”
- 「この日差
( しのなか体育( とか絶対( 無理( だわ…」- “There's no way I can do PE in this blazing sun...”
- “What's with this heat outside...? I'd totally die without the AC...”
Presence and Omission
無事 (Safety)
無事
It combines 無 (without) with 事 (incident) to show being without incident.

After school, Ninako notices Ren sleeping a few seats away on the train. When Ren wakes and rushes for his stop, he knocks Ninako’s phone from her hand. The phone hits the floor, and the strap charm breaks. Ren misses his exit while apologizing and offers to replace the charm.
- 仁奈子
( : - 「だっ 大丈夫
( ですっ」- “I—it's all right.”
- 「ケータイは無事
( だし!」- “Since my cell phone is safe!”
- 「それにこのストラップただのジュースのおまけだし」
- “Furthermore, this strap is just a juice prize.”
- 「しかももうだいぶ前
( 〜〜〜〜のやつで古( いから」- “Plus, it's from way back when, so it's old.”
- “I—it's all right.”