BOYNTON BEACH, Fla. — To turn on the television, open the mail or drive down the highway here is to watch conservatives test the boundaries of how far they can go to disqualify President Obama.

Along the Interstate that connects the beach towns of Florida’s east coast, giant billboards show the president, whom some on the far right have falsely accused of being Muslim, bowing to a Saudi king. Another blares “Stop Obama!” and shows a nuclear warhead with “Iran” painted on it aimed at Israel, a particularly potent message with this area’s many Jewish voters.

In one commercial that started running this month, a wealthy businessman and Hungarian immigrant named Thomas Peterffy laments how socialism shackled his native country. “And that’s what I see happening here,” he says. “That’s why I’m voting Republican and putting this ad on television.”

And a new anti-Obama DVD is dropping into voters’ mailboxes, claiming that the president is the love child of an illicit relationship between his mother, Stanley Ann Dunham, and Frank Marshall Davis, a Communist Party loyalist. The back story of the DVD offers the latest example of how secretive forces outside the presidential campaigns can sweep into battleground states days before the election.