The film, starring Bradley Cooper and Abbie Cornish, held off competition from Gore Verbinski's animation Rango, which took $15.3 million, and Matthew McConaughey's comeback legal drama The Lincoln Lawyer, which entered in third place with an estimated $13.4 million.
Rounding out the top five films was Simon Pegg and Nick Frost's latest outing Paul, which took $13.2 million for its launch, despite topping the UK box office last month.
Next weekend sees the wide release of Zack Snyder's fantasy epic Sucker Punch, as well as the family-oriented Diary Of A Wimpy Kid: Rodrick Rules.
This week's US box office top ten in full:
1. (-) Limitless - $19,000,000
2. (2) Rango - $15,300,000
3. (1) Battle: Los Angeles - $14,600,000
4. (-) The Lincoln Lawyer - $13,400,000
5. (-) Paul - $13,200,000
6. (3) Red Riding Hood - $7,300,000
7. (4) The Adjustment Bureau - $5,900,000
8. (5) Mars Needs Mums - $5,300,000
9. (6) Beastly - $3,300,000
10. (7) Hall Pass - $2,600,000
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