The falling dollar has been an elixir for American companies that have come to rely on sales overseas. For example, Nike, which now gets 60 percent of its sales outside of the United States, said last month that the weaker dollar had increased its worldwide revenue by 3 percent in the past quarter.

But what is giving a lift to American multinationals has become a millstone for European companies competing in the United States.

Take BMW, which said in August that its second-quarter earnings fell 4.6 percent as sales in the United States were eroded by the weaker dollar. The German automaker competes against companies in Japan, whose currency has not risen as much as the euro has, as well as against Detroit. BMW has not recently raised its prices in the United States, but it may have to as its profit margins come under increasing pressure.

With the euro having recently risen to 1.40 to the dollar, the exchange rate has "attained a pain threshold for European companies,'' according to the president of the European business lobbying group, BusinessEurope. Other European currencies have been climbing against the dollar as well. Measured against an index of major-trading-partner currencies, the dollar has reached a 15-year low. Against some currencies, the move is even more extreme; last month the dollar reached parity with the Canadian dollar, known as the loonie, for the first time in 31 years.

Here is the value of the dollar against the euro over the last 12 months:

The problem for European companies is twofold: They must remain competitive with U.S. companies abroad, and they must continue to woo consumers within the U.S, a country that is not only the European Union’s single largest market but also one battling sluggish growth.

''The United States is a huge market and the last thing they want to do is lose too much market share, because it is hard to get it back,'' says Steve Hoch, a marketing professor at the University of Pennsylvania's Wharton School. ''A lot of manufacturers will take it; they’d rather subsidize in the short run than try to go back later and buy back share.”