Long after he had been promoted from Mayor of Houston to president of its chamber of commerce, Louie Welch was asked just how far the sprawling city should continue to spread. Weren't there limits somewhere?

"I don't see where," he replied.

It has been an article of faith here that Houston's almost unlimited power to reach out and annex its affluent new suburbanites, their shopping centers and their industry has kept the city government fiscally sound, its tax base thriving and its tax rate low.

No matter that many annexed residents have not received full city services years after they were brought into the city limits.

No matter that homes have burned down while confused fire dispatchers tried to figure out if a reported blaze was actually in Houston proper.

No matter that the city was forced to pay a man $268,000 after his van flipped over in a cavernous pothole and his arm was cut off.

No matter that many of the municipal services provided by this financially sound government are about as shoddy as those provided by a bankrupt one.

But it may matter now.

For while Houston has had trouble catching up with its annexations, annexation is catching up with Houston. A series of pressures brought on the city and a drawning sense of the limits of rapid growth have now forced Houston to a critical turning point.

The outcome will have enormous implications for the future. What is happening now will determine the city's financial health, as well as whether blacks and Hispanics finally get some share of the power in running America's fifth most populous city. It may even determine whether many of the city's moonscape-like streets ever get repaired.

Facing court actions, a tax protest movement, a coming state legislative session that could review annexations, and U.S. Justice Department action, the city council vastly scaled down its latest annexation effort. The council, fully aware of its inability to properly serve all of the proposed new residents, took in fewer than half the suburbanites it had originally considered.

"We're in a crunch," says Richard Murray, a University of Houston political scientist and student of city affairs. "For many years the city has proceeded without any real concern for minority areas' representation and for newly annexed areas' demand for services."

Now too many people here are, in Murray's word, "irked-off."

So one reason the council pulled back on its annexation was the inability to serve new people.

Indeed, the chief of police had talked about hiring the county sheriff's office to patrol some new areas, so thinly spread is today's force. The city canceled one recent police cadet class for lack of applicants; the mayor earlier had decreed an end to "trash" on the problem-plagued force.

The halcyon days of annexation are over.

Having doubled its area in 1956 alone and having added 200 square miles since, Houston now booms on 521.1 square miles of rather bleak landscape - seven times its size at the end of World War II. Its population is 1.5 million.

A year ago, some 94,000 people, overwhelmingly white and residents of suburban Clear Lake City, were added to the city's population overnight, in what was surely the last unfettered annexation grab here.

The city's black and Hispanic minorities comprise roughly 45 percent of the population, but in 23 years have been able to win only six of 120 city council and administrative posts in at-large, city-wide elections.

The Voting Rights Act of 1975 gives the U.S. Justice Department the power to review any city action here that may affect minorities' ability to vote and elect officials, and last month Justice directed the city to submit the Clear Lakes round of annexations for such review.

It was a move that, combined with a court effort by several minority groups, could lead to the city's abandonment of electing an establishment, white-machine-dominated council under at-large voting. In the past, Justice has pressured localities to use single-member districts after annexations diluted minority voting power by less than the two or three precentage points that the Clear Lakes annexations did here.

And that was before the recent 40,000 mostly white suburbanites were added.

So it appears that Houston is going to have to choose between continued annexations that keep it healthy and radical changes in political power structure, giving city council districts to black and Hispanic areas. This will be an interesting change in what was once a sleepy old southern city where an unelected cabal of the power elite would run the city from a backroom.

Annexation is "crucial to the economic vitality of cities," says Susan A. MacManus, another University of Houston political scientist, who, with Murray, has studied 243 large American cities.

Not only does it add to the real estate tax base, but it increases payments from the state based on population.

But annexation, as practiced here, is also a clever form of short-term borrowing in a city that vocally preaches "pay-as-you-go." Houston quickly gets new revenues but may take months or years to gear up fire departments, police patrols and ambulance service to serve its new areas.

Or you just spread services a little thinner for everyone.

THIS DOES, OF COURSE, KEEP TAX RATES DOWN AND HELP GIVE Houston its image of being favorable to business, even though it irks people off.

Crime is overwhelming the police force, and last August residents approved a sales tax increase to finance a modern bus system, so terrible has traffic congestion become. But they were told recently it will take two years to get adequate bus service.

This was bad news for commuters who, the Census Bureau has now disclosed, drive farther to work than workers in any other metropolitan area and who, on average, take only seven fewer minutes to get to work than... New Yorkers.

The traffic has resulted in such pleasantries as a man striking up an acquaintance with a woman in the car alongside - and pulling off to weather the rush hour in a cocktail lounge. More often, though, the streets are not so enjoyable.

Fully one-third of the people here are irked-off at street conditions or noise from city streets, the Census Bureau says. In some places, rather than fix the streets the city just puts up signs warning "dip" or "bump."

"The incredible rapid expansion is over," says Barry J. Kaplan, a University of Houston urban historian. "All of a sudden the council is having second thoughts. There is an awareness of limits."

"We're up to the point," says Councilman Larry McKaskle, "where if we take on more we will be overextended completely." Some would say that point was reached years ago.

But it is clear that, should Houston have to decide between continued annexations and a new council system. there will still be annexations. "Annexations and a new council system, state law, but there will be political change," offers Mayor Jim McConn.

"Houston has to annex, period," Kaplan says "Migration here has always been outward. It may not be good for the people being annexed, but without it, Houston will be another Cleveland."

And McKaskle sees a population in Houston in 100 years of 28 million people, adding, "We have not seen any growth in Houston."

But even Louie Welch is now talking about "quality growth."