An example of the city's new approach to ''managed'' growth is Gary L. Bradley's plans to develop the old Circle C Ranch, 3,600 acres of cedars and live oaks southwest of the city that is in the aquifer area and outside Austin's ''preferred'' growth corridor. Mr. Bradley, a 34-year-old West Texas native who has been in Austin since 1968, is negotiating with the city to provide him water and sewer lines by approving a municipal utility district with authority to issue bonds for his project, which would ultimately have 7,000 homes and apartments.

In exchange, Mr. Bradley has offered to build special retention dams to reduce runoff pollution and to limit paving and density.

''The city does not have to extend utilities to me,'' he said. ''But they want to because they do not want me to buy water from the river authority.'' Moreover, he said, he is cognizant of what he affectionately calls the ''granola army,'' environmentalists who ''can beat you without money.''

''We've got a town with a conscience,'' he said. ''We will not have another Houston. We have too many safeguards.''

Others are less hoepful. ''I don't see any way of avoiding the fate that awaits us,'' said Kenneth Manning, a 38-year-old lawyer and environmental leader who used to work for Mr. Bradley. He said the city was unable to take a strong hand in channeling development because ''it is extremely difficult to get the City Council to tell a developer 'no' once in a while.'' All six Council members and the new Mayor ran with contributions from developers in April's elections. The Best of All Worlds

Austin in a way has the best of all worlds: the fine restaurants, theaters and good bookstores of urban life, yet a small-city layout with lots of parks that lets you get home from work in 15 minutes. Many of its residents are Texans who came to study at the university and stayed, many of them professionals who have sacrificed more lucrative careers elsewhere. Many artists, writers, poets and artisans have also gravitated here.

It is just these things that have brought high-technology businesses seeking refuge from the high costs and congestion in California's high-technology area and wanting an agreeable setting to help recruit staff. Austin's population swelled from 254,000 in 1970 to 345,000 in 1980. The chamber estimates its has since grown to more than 367,000, and some estimates say the metropolitan area will exceed a million by the year 2000. Over the last decade, the number of passengers using the municipal airport has grown from 600,000 to more than 2 million yearly. The growth is accelerating. Since 1979 2.6 million square feet of office space has been built; 2.3 million more is now under construction.