Absence and nothingness with

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

(Incompetent)

無能(むのう) 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.

Manga panel from 名探偵コナン showing example of An incompetent officer with 無能.
名探偵コナン » Volume 7 » Page 8

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(ささい):
「名探偵(めいたんてい)のあなたからもいってやってくださいよ。 この刑事(けいじ)がいかに無能(むのう)かという事(こと)を…」
“You, the great detective, should tell him yourself. Just how incompetent this police detective is...”
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Limits and Possibility

(Impossible)

無理(むり) means impossible, unreasonable, or too demanding. In conversation, it can mean “that’s impossible” or warn someone not to push too hard.

It combines 無 (without) with 理 (reason) to show something without reasoning.

Example 1 of 2: I'd totally die in the heat with 〜じゃう
Example 1 of 2
Manga panel from ホリミヤ showing example of I'd totally die in the heat with 〜じゃう.
ホリミヤ » Volume 1 » Page 45

On a summer day, Hori and her friends in class can’t ignore the heat.

由紀(ゆき):
「外(そと)の暑(あつ)さってなんなの…? クーラーないと死(し)んじゃうって…っ」
“What's with this heat outside...? I'd totally die without the AC...”
「この日差(ひざ)しのなか体育(たいいく)とか絶対(ぜったい)無理(むり)だわ…」
“There's no way I can do PE in this blazing sun...”
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Example 2 of 2: Not moving while staring with 〜ながら
Example 2 of 2
Manga panel from ご注文はうさぎですか? showing example of Not moving while staring with 〜ながら.
ご注文はうさぎですか? » Volume 1 » Page 113

Cocoa buys Chino a puzzle to make up for finishing the one Chino had been working on. The two and their friends work at putting the new puzzle together.

チノ:
「ココアさんさっきからピース見(み)つめながら動(うご)かない…」
“Cocoa-san has been staring at the puzzle piece while not moving for a while now...”
「その…責任(せきにん)取(と)ろうと無理(むり)しないでください」
“Um... Please don't push yourself trying to take responsibility.”
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Presence and Omission

(Safety)

無事(ぶじ) means safe, unharmed, or without incident.

It combines 無 (without) with 事 (incident) to show being without incident.

Manga panel from ストロボ・エッジ showing example of Furthermore, it's just a bonus item with それに〜.
ストロボ・エッジ » Volume 1 » Page 23

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(になこ):
「だっ 大丈夫(だいじょうぶ)ですっ」
“I—it's all right.”
「ケータイは無事(ぶじ)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(まえ)〜〜〜〜のやつで古(ふる)いから」
“Plus, it's from way back when, so it's old.”
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(Without permission)

無断(むだん) means without permission or prior notice.

It combines 無 (without) with 断 (decision) to show something done without approval.

Manga panel from くノ一ツバキの胸の内 showing example of Doing as you please is strictly forbidden with 〜べからず.
くノ一ツバキの胸の内 » Volume 1 » Page 106

Sazanka sneaks out of class, leaving the teacher to send Asagao to bring her back. As Asagao chases Sazanka across the rooftops near the upper grade classroom, Tsubaki spots them. She meets them up on the roof and recites the school’s first rule.

ツバキ:
「ひとつ。無断(むだん)で勝手(かって)な行動(こうどう)をとるべからず。」
“One: Doing as you please without permission is strictly forbidden.”
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