The spectacle of it alone is alluring: one of the world’s most famous women onstage for 90 commercial-free minutes alongside one of the country’s most recognizable men, in the ultimate convergence of celebrity and politics.

But with a presidential race that once seemed to be tipping in Mrs. Clinton’s favor growing more competitive as early voting begins, the debate at Hofstra University in New York is far more than a made-for-TV moment.

National polls and swing state surveys could hardly be tighter heading into the first presidential debate.

Quinnipiac University declared the race “too close to call” on Monday, as its latest national poll of likely voters found Mrs. Clinton edging Mr. Trump by a margin of 47 percent to 46 percent in a head-to-head matchup.

A separate poll of likely voters from Monmouth University showed Mrs. Clinton holding a lead of four-percentage points in a four-way contest, leading by a margin of 46 percent to 44 percent. A month ago, Mrs. Clinton led by 7 points.

Polls conducted CNN/ORC in Colorado and Pennsylvania that included Gary Johnson and Jill Stein, the Libertarian and Green Party candidates, also offered little clarity. Mr. Trump holds a one percentage point advantage in Colorado, while Mrs. Clinton leads by the same margin in Pennsylvania.

In both states, voters with college degrees overwhelmingly support Mrs. Clinton. Those with less education heavily favor Mr. Trump.

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Here is what else to look for as the candidates try to motivate — or reassure — supporters and win over a small group of undecided voters.

Can Trump demonstrate he’s fit for the Oval Office?

With a showman’s flair for generating publicity and a firebrand’s talent for touching the rawest of nerves, Mr. Trump has effectively harnessed the angst of many in the country who are not feeling the effects of the economic upswing, have tired of prolonged wars or have grievances about a changing country.

But surveys show that a majority of Americans still believe he is unqualified to be president. If Mr. Trump is to convince those voters who have doubts about his fitness for high office, but are uneasy with Mrs. Clinton, the three debates represent his best opportunity to prove he can be trusted to serve as a head of state. He has to show discipline when it comes to how he engages Mrs. Clinton, challenging her without belittling her, and at least show he is conversant on foreign and domestic policy issues.

Just as Ronald Reagan did in 1980 and George W. Bush did in 2000, Mr. Trump faces a threshold test about whether he is up to the job. But neither Mr. Reagan nor Mr. Bush faced the same level of doubt as Mr. Trump does today. The provocateur must show he can be a president.