Why didn't Batman just knock down the Joker while he had the chance?

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In the early stages of Batman Arkham Asylum, after Batman defeats the giant mutant prisoner, the Joker gives him a chance to take him down for good:
Although it was probably the Joker's intent, it seems Batman saw something in that particular scene that made him withdraw.
Knowing the Batman, I'm pretty sure he could have come up with a plan that would have trapped the Joker somehow and stopped his carnage in the asylum - ending all of the game story right there.
But why did Batman stop before taking down the Joker?
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Is there a need for Batman to kill the Joker?
If Batman killed the Joker, the people of Gotham would feel much safer knowing a criminal, who has killed innocent lives, is dead. The city of Gotham would finally be at peace, without having to look over their shoulders. If Batman did kill the Joker, it might have an effect on not only him but Bruce Wayne.Why did Batman save the Joker?
In the next movie, Batman saved the Joker because he threw him off. While in this game, it was Bane who blasted the Joker off the building, but Batman was being very suicidal to save Joker's life. (And I wonder why Joker didn't choose to flee but wanted to enjoy as Bane fired a rocket at him.)Why Didn't You Stop Me?
Sources: Stack Exchange - This article follows the attribution requirements of Stack Exchange and is licensed under CC BY-SA 3.0.
Images: Anna Shvets, John Barnard, RODNAE Productions, Ali Pazani