“One of the reasons why you’re seeing more strikes there is because there’s more ISIL there,” Rear Adm. John F. Kirby, the Pentagon press secretary, said Wednesday, using an acronym for the Islamic State. He said that “hundreds” of militants have been killed.

But Kobane also has come to represent a potential propaganda victory the Obama administration is eager to deny the militants.

“Part of the dynamic we want to show is that these guys aren’t ten feet tall,” said a senior administration official who spoke on the condition of anonymity to discuss the motivation for target selection beyond purely military objectives. “A lot of their edge has been psychological.”

“They’re like a shark; if they’re not swimming, they’re sinking. That’s how they recruit foreign fighters and establish themselves as the vanguard of global jihad,” the official said.