In what ways does Hrothgar live up to or fail to live up to the image of the epic hero?
I would disagree with the above answer; Hrothgar is an epic hero, but not the hero of this particular epic. The following qualities define an epic hero.
1. A person of high or noble stature. -- Hrothgar is king.
2. A person who is brave and strong. --While currently weak because of his age, he once possessed all of these qualities.
3. Has intelligence and moral qualities that the people of his culture respect. --Hrothgar is a kind and God-loving king; his people do love him.
4. The hero has a flaw. --Much like many characters in epics, Hrothgar's flaw is his "hubris," or excessive pride (in Greek) and desire for wealth. In fact, he fears that Grendel is a punishment from God because of his obsession with power and money.
5. Is look favorably upon by deities, or is part-deity. --Though he believes Grendel to be a punishment from God, because Hrothgar does love and strongly believe in God, he believes Grendel personally spares him.
Hrothgar fits all the characteristics of an epic hero, though is clearly not the main protagonist of the tale of Beowulf. One may argue that Hrothgar is actually a foreshadowing of what Beowulf will become. While loved by his people, and surely a good king, Beowulf eventually succumbs to his pride, his age, and his desire for wealth.
What an interesting question, not one I had before considered. Hrothgar is the second most important character in Beowulf--a sure sign that he is not the epic hero of this tale. I guess the easiest answer is that Hrothgar is not an epic hero because he allows Grendel to become a marauder of men (and women and children, presumably) in his keeping for so many years. If he had found a way, either for himself or someone else, to get rid of Grendel, then he would clearly have been an epic hero. But that doesn't happen. Instead, he waits rather helplessly for someone else to come to his people's rescue. Thankfully, Beowulf arrives.
Hrothgar will have to settle on the title of epic host. Once Beowulf arrives, Hrothgar treats him with courtesy and respect, honoring his presence before Grendel's death and bestowing honor upon him after he kills the invader. He is gracious and generous, but he is not an epic hero. That honor goes to Beowulf.