There was the time last winter when President Biden was awakened at 3 a.m. while on a trip to Asia and told that a missile had struck Poland, touching off a panic that Russia might have expanded the war in Ukraine to a NATO ally. Within hours in the middle of the night, Mr. Biden consulted his top advisers, called the president of Poland and the NATO secretary general, and gathered fellow world leaders to deal with the crisis.And then there was the time a few weeks ago when the president was hosting children for Take Your Child to Work Day and became mixed up as he tried to list his grandchildren. “So, let me see. I got one in New York, two in Philadelphia — or is it three? No, three, because I got one granddaughter who is — I don’t know. You’re confusing me.” He also drew a blank when asked the last country he had visited and the name of a favorite movie.The two Joe Bidens coexist in the same octogenarian president: Sharp and wise at critical moments, the product of decades of seasoning, able to rise to the occasion even in the dead of night to confront a dangerous world. Yet a little slower, a little softer, a little harder of hearing, a little more tentative in his walk, a little more prone to occasional lapses of memory in ways that feel familiar to anyone who has reached their ninth decade or has a parent who has.The complicated reality of America’s oldest president was encapsulated on Thursday as Congress approved a bipartisan deal he brokered to avoid a national default. Even Speaker Kevin McCarthy testified that Mr. Biden had been “very professional, very smart, very tough” during their talks. Yet just before the voting got underway, Mr. Biden tripped over a sandbag at the Air Force Academy commencement, plunging to the ground. The video went viral, his supporters cringed and his critics pounced.Anyone can trip at any age, but for an 80-year-old president, it inevitably raises unwelcome questions. If it were anyone else, the signs of age might not be notable. But Mr. Biden is the chief executive of the world’s most powerful nation and has just embarked on a campaign asking voters to keep him in the White House until age 86, drawing more attention to an issue that polls show troubles most Americans and is the source of enormous anxiety among party leaders.“You say I’m ancient?” Mr. Biden said at the White House Correspondents’ Association dinner in April. “I say I’m wise.”Yuri Gripas for The New York TimesThe portrait that emerges from months of interviews with dozens of current and former officials and others who have spent time with him lies somewhere between the partisan cartoon of an addled and easily manipulated fogy promoted by Republicans and the image spread by his staff of a president in aviator shades commanding the world stage and governing with vigor.It is one of a man who has slowed with age in ways that are more pronounced than just the graying hair common to most recent presidents during their time in office. Mr. Biden sometimes mangles his words and looks older than he used to because of his stiff gait and thinning voice.Yet people who deal with him regularly, including some of his adversaries, say he remains sharp and commanding in private meetings. Diplomats share stories of trips to places like Ukraine, Japan, Egypt, Cambodia and Indonesia in which he often outlasts younger colleagues. Democratic lawmakers point to a long list of accomplishments as proof that he still gets the job done.His verbal miscues are nothing new, friends note; he has struggled throughout his life with a stutter and was a “gaffe machine,” to use his own term, long before he entered Social Security years. Advisers said his judgment is as good as ever. So many of them use the phrase “sharp as a tack” to describe him that it has become something of a mantra.Mr. Biden says age is a legitimate issue but maintains that his longevity is an asset, not a liability. “You say I’m ancient?” he said at the White House Correspondents’ Association dinner in April. “I say I’m wise.”Still, few people fail to notice the changes in one of the nation’s most public people. As vice president a dozen years ago, Mr. Biden engaged in energetic squirt gun battles each summer with the children of aides and reporters. More than a decade later, he shuffled stiffly across the Edmund Pettus Bridge in Selma, Ala., to mark the anniversary of Bloody Sunday.Polls indicate the president’s age is a top concern of Americans, including Democrats. During a recent New York Times focus group, several voters who supported Mr. Biden in 2020 expressed worry, with one saying: “I’ve just seen the blank stare at times, when he’s either giving a speech or addressing a crowd. It seems like he loses his train of thought.”Unease about Mr. Biden’s age suffuses Democratic circles. One prominent Wall Street Democrat, who like others spoke on the condition of anonymity to avoid offending the White House, noted that among party donors it was all anyone was talking about. At a small dinner earlier this year of former Democratic senators and governors, all of them in Mr. Biden’s generation, everyone at the table agreed he was too old to run again. Local leaders often call the White House to inquire about his health.In private, officials acknowledge that they make what they consider reasonable accommodations not to physically tax an aging president. His staff schedules most of his public appearances between noon and 4 p.m. and leave him alone on weekends as much as possible.A study of Mr. Biden’s schedule based on data compiled by Axios and expanded by The New York Times found that Mr. Biden has a similar morning cadence as the president he served, Barack Obama. Neither had many public events before 10 a.m., just 4 percent in Mr. Obama’s last year in office and 5 percent in Mr. Biden’s first two and a half years. But the real difference came in the evening. Mr. Obama was twice as likely to do public events after 6 p.m. compared with Mr. Biden, 17 percent to 9 percent.Aides limit exposing the president to news media interviews when he could make a politically damaging mistake. He has given just a fourth of the interviews Donald J. Trump did in the same time period and a fifth of Mr. Obama’s interviews — and none at all to reporters from a major newspaper. Mr. Biden has not given an interview to the news department of The Times, unlike every president since at least Franklin D. Roosevelt other than Dwight D. Eisenhower. And in the past 100 years, only Ronald Reagan and Richard M. Nixon have subjected themselves to as few news conferences.White House officials have not made Mr. Biden’s doctor available for questioning, as previous presidents have. In February, Kevin C. O’Connor, the White House physician, issued a five-page letter stating that Mr. Biden is “fit for duty, and fully executes all of his responsibilities without any exemptions or accommodations.”But he also wrote that the president’s tendency to walk stiffly is “in fact a result of degenerative (‘wear and tear’)” changes in his spine, and partly the result of “tighter hamstrings and calves.” The letter said there were “no findings which would be consistent with” a neurological disorder like stroke, multiple sclerosis or Parkinson’s disease. He takes medicine for atrial fibrillation, cholesterol, heartburn, asthma and allergies.Like many his age, Mr. Biden repeats phrases and retells the same hoary, often fact-challenged stories again and again. He can be quirky; when children visit, he may randomly pull a book of William Butler Yeats off his desk and start reading Irish poetry to them.At the same time, he is trim and fit, exercises five days a week and does not drink. He has at times exhibited striking stamina, such as when he flew to Poland then boarded a nine-hour train ride to make a secret visit to Kyiv, spent hours on the ground, then endured another nine-hour train ride and a flight to Warsaw. A study of his schedule by Mr. Biden’s aides shows that he has traveled slightly more in the first few months of his third year in office than Mr. Obama did in his.Mr. Biden’s trip to Kyiv, in which he met with President Volodymyr Zelensky of Ukraine, required a nonstop schedule.Daniel Berehulak/The New York Times“Does he ramble? Yes, he does,” said Gov. Phil Murphy of New Jersey, a Democrat who categorically rejects the idea that Mr. Biden is too old to be president. “Has he always rambled? Yes, he has. Public and private. He’s the same guy. He’s literally — I’m not saying this lightly. I don’t know anyone else in my life who is so much the same guy privately as he is publicly.”Some friends bristle at the attention to his age. “I think the reason this is an issue is primarily because of the media talking about it constantly,” said former Senator Ted Kaufman, a longtime adviser to Mr. Biden from Delaware. “I do not see anything in my dealings with him that age is a problem. He’s done more than any president has been able to do in my lifetime.”Andrew Bates, a White House spokesman, noted that Republican hard-liners were grousing that Mr. Biden had gotten the better of Mr. McCarthy in the fiscal deal. “It’s telling that the same extreme MAGA members of Congress who’ve been talking about his age complained this week that he outsmarted them on the budget agreement,” Mr. Bates said.The question of Mr. Biden’s age does not come in isolation, of course. Mr. Trump, his likeliest Republican challenger, is just four years younger and was the oldest president in history until Mr. Biden succeeded him. If Mr. Trump were to win next year, he would be 82 at the end of his term, older than Mr. Biden will be at the end of this one.While in office, Mr. Trump generated concerns about his mental acuity and physical condition. He did not exercise, his diet leaned heavily on cheeseburgers and steak and he officially tipped the scales at 244 pounds, a weight formally deemed obese for his height.After complaining that he was overscheduled with morning meetings, Mr. Trump stopped showing up at the Oval Office until 11 or 11:30 a.m. each day, staying in the residence to watch television, make phone calls or send out incendiary tweets. During an appearance at the U.S. Military Academy at West Point, he had trouble lifting a glass of water and seemed to have trouble making his way down a modest ramp.Most striking was Mr. Trump’s cognitive performance. He was erratic and tended to ramble; experts found that he had grown less articulate and that his vocabulary had shrunk since his younger days. Aides said privately that Mr. Trump had trouble processing information and distinguishing fact from fiction. His second chief of staff, John F. Kelly, bought a book analyzing Mr. Trump’s psychological health to understand him better, and several cabinet secretaries concerned that he might be mentally unfit discussed invoking the 25th Amendment to remove him.Former President Donald J. Trump’s cognitive issues are not as often associated with age in the public mind as are Mr. Biden’s, perhaps because Mr. Trump’s bombastic volume conveys energy.Doug Mills/The New York TimesBut perhaps because his bombastic volume conveys energy, Mr. Trump’s issues are not associated with age in the public mind as much as Mr. Biden’s are. In a recent Reuters/Ipsos poll, 73 percent said Mr. Biden is too old to be in office, compared with 51 percent who said the same of Mr. Trump.Mr. Biden manages his day with more discipline than his predecessor. Jill Biden, who teaches at Northern Virginia Community College, gets up around 6 a.m. while the president wakes an hour later, according to accounts he has given. Mr. Biden has told aides that their cat sometimes wakes him in the middle of the night by walking across his face.By 7:20 a.m., the first lady leaves for work. Mr. Biden works out at 8 a.m.; he has a Peloton bicycle in the residence and is known to watch shows like “Morning Joe” on MSNBC. He arrives at the Oval Office by 9 a.m. for a morning usually filled with meetings. For lunch, there is a rotation of salad, soup and sandwiches.Mr. Biden exercises five days a week and does not drink.Al Drago for The New York TimesFollowing afternoon events, the president returns to the residence around 6:45 p.m. For dinner, pasta is a favorite. In fact, one former official said, whenever he travels, aides make sure there is always red sauce on hand for pasta to finish his day — even as he balks at the salmon that his wife urges on him.From 8 p.m., the Bidens often read their briefing books together in the living room of the residence. The first lady typically turns in at 10:30 p.m. and the president follows a half-hour later.Aides say it is clear he actually reads the briefing books because of the questions that follow. “There’s no one who is better at asking questions to get to the bottom of an issue, calling your bluff, asking the tough questions,” said Stefanie Feldman, the White House staff secretary. “He asks just as tough questions today as he did 10 years ago.”Some who accompany him overseas express astonishment at his ability to keep up. When Italy’s new leader pushed for a meeting while the president was in Poland, he readily agreed to add it to the already packed schedule. During a trip to Ireland, people with him said he was energized and wanted to talk at length on Air Force One rather than rest.Still, after fatiguing days on the road, he skipped dinner with world leaders in Indonesia last year and again in Japan in May. Others who have known him for years said privately that they have noticed small changes. When he sits down, one former official said, he usually places a hand on his desk to hold his weight and rarely springs back up with his old energy.Mr. Biden’s staff schedules most of his public appearances between noon and 4 p.m. and leaves him alone on weekends as much as possible.Doug Mills/The New York TimesHe speaks so softly that he can be hard to hear. For speeches, aides give him a hand-held microphone to hold close to his mouth to amplify his voice even when standing at a lectern with mounted microphones.Mr. Biden and Jill Biden, his wife, often follow a similar schedule.Doug Mills/The New York TimesYet aides said that while he can momentarily forget a name or fact, he retains a formidable memory for detail. Preparing to travel to Shanksville, Pa., on the 20th anniversary of the attacks of Sept. 11, 2001, he became frustrated that officials had given him the wrong plan for his movements. He had been to the memorial before and knew the plan made no sense because he remembered the layout of the grounds.White House officials voice aggravation that concern about age is inflated by pictures on the internet that are sometimes faked or highly distorted. Every week, strategists conduct a word cloud analysis with a panel of voters asking what they had heard about the president, good or bad. After Mr. Biden’s foot got caught in the toe cage of his bicycle and he tumbled over last year, the two words in the bad-word cloud for weeks were “bike fell” — all the more frustrating for aides who noted that Mr. Trump hardly seemed capable of even riding a bike.Mr. Biden lately has turned to self-deprecating humor to defuse the issue, taking a cue from Mr. Reagan, who won re-election in 1984 at age 73 in part with a well-timed debate quip about not exploiting “my opponent’s youth and inexperience.”Some who accompany Mr. Biden overseas express astonishment at his ability to keep up. Still, after fatiguing days on the road, he skipped dinner with world leaders in Indonesia last year and again in Japan in May.Doug Mills/The New York TimesAt the correspondents’ dinner, Mr. Biden assured the audience that he supported the First Amendment, and “not just because my good friend Jimmy Madison wrote it.” During the Take Your Child to Work Day event, he looked back on “when I was younger, 120 years ago.”And at the Air Force Academy a few days ago, Mr. Biden joked that “when I was graduating from high school 300 years ago, I applied to the Naval Academy.” After tripping on the sandbag, he sought to laugh that off too. “I got sandbagged,” he said. More