Mexico City, Mexico — President Obama, in speech to high school and university students in Mexico's capital last Friday, said that it was time to banish the stereotypical Mexico of violence and people fleeing across borders and embrace the new image of a strengthening democracy and economy.

"I have come to Mexico because it is time to put old mind-sets aside," Obama said to vigorous applause from hundreds of students at the National Anthropology Museum. "It’s time to recognize new realities, including the impressive progress in today’s Mexico. For even as Mexicans continue to make courageous sacrifices for the security of your country, even as Mexicans in the countryside and in neighborhoods not far from here struggle to give their children a better life, it’s also clear that a new Mexico is emerging."

Although poverty remains deep and wages have stagnated, Mr. Obama focused on the positive signs of the economy, including growth measurements that exceed those in the United States, a surge in the manufacturing and technology industries, and rising levels of middle class Mexicans.

The battle in Mexico against drug trafficking and organized crime, in which the US has invested more than $1 billion in recent years with mixed results, has dominated the agenda between the nations, during the Obama and George W. Bush administrations.

After a dinner Thursday night with President Enrique Peña Nieto, Mr. Obama told the gathering on Friday "we agree that the relationship between our nations must be defined — not by the threats we face — but by the prosperity and opportunity we can create together. And if we are serious about being equal partners, then both our nations must recognize our mutual responsibilities."

Some American officials have grown worried that Mexico has pulled back on the deep involvement of the United States — developing and passing along key intelligence and deploying spy technology on their territory — in drug investigations.

But President Obama hardly mentioned the security issues in his speech and remarks after arriving on Thursday, in keeping with Mexico’s wishes to define the relationship more on the economy and less-thornier subjects.

Whatever disputes may be playing out behind the scenes between the governments, the visit was an occasion to bask in the public’s adulation, with large crowds craning over barricades to catch a glimpse of him or his motorcade. The Mexican news media have given the visit wall-to-wall coverage, and Friday morning newspaper front pages were filled with pictures of Mr. Obama and Mr. Peña Nieto grinning and strolling.

Mr. Obama, seeking a connection with the young audience, sprinkled his remarks with the little Spanish he knows — "juntos, podemos lograr más" — "together, we can achieve more," he said at one point to the delight of the crowd.

"In you, Mexico’s youth, I see a generation empowered by technology. I think I see some of you tweeting and WhatsApping right now," he said to laughter.

But more than anything, Mr. Obama, who later met privately with business leaders at the museum, sought to deliver a message of hope that stronger economic ties and trade could only benefit both countries, reduce crime and the desire by Mexicans to emigrate to the United States.

"I also believe that the long-term solution to the challenge of illegal immigration — so we’re not dealing with this, decade after decade — is a growing, prosperous Mexico that creates more jobs and opportunity for young people right here," he said.

He made sure to acknowledge the role of the United States in Mexico’s problems, particularly insecurity.

"We understand that the root cause of much of the violence here — and so much suffering for many Mexicans — is the demand for illegal drugs, including in the United States," Mr. Obama said. He also said most of the guns that are used to commit violence in Mexico come from the United States.

"I will continue to do everything in my power to pass common sense reforms that keep guns out of the hands of criminals and dangerous people that will save lives in both our countries," he said to applause.