What was going on when I was born?

For years Dome Petroleum Ltd. was a company that glowed. In the increasingly lucrative oil and natural gas business, it was among the pioneers developing new drilling techniques in the forbidding Beaufort Sea. The stock market applauded, pushing the company's shares to new highs, while people happily referred to its chairman and founder as ''Smiling Jack.'' Today Dome Petroleum is not only in trouble but also is tarnishing the Canadian Government's National Energy Program and making five of the country's major banks extremely edgy about the losses they are risking. The company has historically relied on heavy borrowing to finance its growth, but its debts now are staggering and coming due fast. By the end of 1981, Dome Petroleum's long-term debts were bigger than those of the Ford Motor Company and the Chrysler Corporation combined. As of March 31, Dome reported total borrowing of $5.8 billion (Canadian money) and no cash or short-term deposits whatsoever. A Canadian dollar is now worth about 78 United States cents.

International A fierce artillery fight in west Beirut seemed to indicate that the Palestinian guerrillas besieged by Israeli armored units still had a great deal of ammunition, weapons and resolve. For the first time, the Palestinians fired back on Israeli gun positions in Christian east Beirut. The battle broke up negotiations for a settlement as all diplomatic efforts focused on a cease-fire. (Page A1, Column 6.) Philip C. Habib needs more time to bring about an agreement in Beirut for the withdrawal of Palestinian forces, Defense Secretary Caspar W. Weinberger said. ''A realistic assessment'' of the time needed by Mr. Habib, the special American envoy, is three more weeks, Mr. Weinberger said. Other Administration officials said, however, that they were far from certain that Israel would continue to hold off an invasion of west Beirut for that long. (A6:3-6.)

On the ears of strollers and joggers, bicyclists and motorists, headphones are appearing in force here, and the town fathers have decided that enough is enough. Woodbridge officials, saying they are the first in the nation to do so, have voted an ordinance to limit the wearing of headphones on the town's streets for safety reasons. If the state approves it, the wearing of headphones will be allowed on sidewalks, but when a pedestrian steps over the curb he will have to leave the world of his tiny tape player or radio and return to the sounds of the street. The ordinance also applies to people riding bicycles and driving cars.

Israel is in a quandary about how to get the Palestine Liberation Organization out of Beirut. The political negotiations seem less promising with each passing day. The military option, with its inevitable bloodshed, seems to grow more complex, more costly and less attractive as time drags on. But it also grows more likely, especially as rocket and mortar duels between Palestinian and Israeli forces increase daily. ''The P.L.O. every day raises new conditions and demands,'' said one official in Jerusalem. ''We get nowhere. We are turning around in a circle. We are going backwards instead of forwards. I think it is only a matter of days until we give up this approach.''

One month after its forces surrendered in Stanley, Argentina is a changed nation searching for a new direction. The defeat in the Falklands combined with an economic recession, three-digit inflation, the imposition of Government control over a free-market economy, political confusion, international isolation and even the loss in the World Cup last week have left many Argentines demoralized about their past and questioning the future. A mood of uncertainty hangs over the country. ''I feel dizzy,'' a businessman said. ''All the traumatic shocks came together at once.'' Gone are the signs of nationalistic euphoria that existed during the war -the rousing television advertisements, the billboards declaring ''Las Malvinas son Argentinas,'' using the Argentine name for the Falklands.

First of three articles. By PAUL GOLDBERGER With huge new office towers filling midtown, with housing demand forcing the conversion of factories and warehouses, and with vast public projects finally getting under way, development is changing the economic and cultural heart of New York City into something physically denser than ever before. Experts differ over what lies ahead for the city's core - the residential and business centers of Manhattan south of 96th Street, where virtually all of the borough's growth is taking place. Will it be a fresher, more efficient city, full of the amenities new architecture can provide? Or will it be more crowded, more pressured and ultimately less comfortable? And will the kind of growth and change that is occurring continue? For midtown Manhattan, greater concentration, greater congestion and more high-rise construction seem virtually certain. Despite the current economic downturn and new zoning regulations designed to limit development in the midtown's heavily developed East Side, so many projects are already in process that the result will be a significant increase in density by the middle of the 1980's, even if no additional building projects at all are initiated.

Seven years after municipal workers gave up part or all of a pay increase to help save New York City from default at the onset of its fiscal crisis, the Koch administration has indicated that it is now willing to pay the money. Officially, no deal has been made and many problems remain, city negotiators said. But the two sides are close enough that Victor Gotbaum, the municipal labor leader, said yesterday that there was ''an understanding'' that the money would be paid. Koch administration officials, not wanting to weaken their bargaining position, said no agreement would be final until all the issues in current contract talks had been resolved.

President Reagan returned from an 11-day California vacation today, with political strategists saying he needs quick, clear signs of economic improvement to protect the Republican Party from election losses this fall. They say the potential losses could be serious enough to jeopardize his precarious political coalition in Congress over the next two years. To most people, the fall elections still seem far off, but Republican and Democratic strategists believe the economy has reached a critical period for the outcome of the election. Because experience indicates a lag of three or four months until an economic improvement has a broad impact on voters, Democratic poll takers like Peter Hart say it is already too late for the Republicans. Republican pollsters like Robert Teeter and even White House officials assert that Republicans need a clear improvement in the next month or six weeks to have any political benefits.

In his 26 years with the Toyo Kogyo Company, Mitsuji Muraoka has been an accountant, a plant designer, an efficiency watchdog, a materials-handling specialist and a computer systems manager. The zigzagging career that Mr. Muraoka, a 50-year-old division manager, has experienced at Toyo Kogyo, the producer of Mazda cars and Japan's third-largest auto maker, is not unusual in Japan. A management employee in a large Japanese company will typically move into new jobs in different fields repeatedly during his career, at intervals ranging from two to seven years. American and European corporations also have programs of job rotation to broaden the experience and training of their managers. But the Japanese-style shuffling of people into new disciplines every few years is a practice that is rare in Western corporations.

CARL HUBBELL remembers when they called it the ''midsummer classic,'' back in 1933 when the teams were picked by John McGraw and Connie Mack and when Babe Ruth hit the home run that won the first All-Star Game. But he thinks it has changed. ''The All-Star Game started in the Depression, when people needed a job,'' Hubbell said. ''If you had a job, you'd hold onto it; you might not get another one. That's all they thought about in baseball, too. And making the All-Star team helped you keep your job. ''Today, it's a show, more of a television show. The players are the actors. They've got their agents and lawyers, and they've got TV. And they've got so many things to do, baseball gets slighted. They've only got one guy who plays like he was hungry to keep his job: Pete Rose.''

THEY call him ''Auntie'' because of his mature behavior at the age of 18 years old. Giuseppe Bergomi was the youngest player on the field yesterday in the most important soccer game in the world, the final of the World Cup. Italy won, 3-1, to become the champions of the world for the third time as Bergomi and a substitute, Alessandro Altobelli, made mighty contributions. Bergomi shut down Karl-Heinz Rummenigge while Altobelli scored the third goal for Italy after replacing the Francesco Graziani after only seven minutes.

Israeli armored units fought a fierce artillery and rocket duel today with Palestinian guerrillas trapped inside west Beirut. For the first time in the battle of Beirut the Palestinians fired back on a wide range of Israeli gun positions in Christian east Beirut. The fighting appeared to intensify after reports that the Israeli Cabinet was studying military options because the negotiations with the Palestine Liberation Organization were at an impasse. (Page A6.)