Green’s Dictionary of Slang

(US Und.) constr. with the, the maximum sentence; the death sentence.

1924		G. Henderson Keys to Crookdom 410: Limit. Maximum sentence.
1927		C. Coe Me – Gangster 40: Coky had gone to the chair [...] a man that kills a cop is crazy and ought to get the limit.
1930		in C. Carr Our Town (2006) 109: Don’t make this mistake. I’ll give ’em the limit.