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West Ham are flying high in the Premier League, sitting fourth in the table after upstaging 10-man Swansea City 3-1 at Upton Park on Sunday.
Towering English striker Carroll was on target twice as West Ham won their third consecutive league match to move within nine points of leaders Chelsea.
"I think there is no limit, we can go all the way to the top," the 25-year-old said.
"We have a great squad and I can't say we are going to give up, now we are in the top four. We have got a great team and we can go all the way.
"I think with the team we have got and the manager [Sam Allardyce], the way he is playing and coaching everyone, it is fantastic and I don't think we will be giving up at all."
