Analyze the role of the heroic ideal and heroic paradox in Beowulf.

I'm not entirely sure what you mean by "the heroic paradox," and
even the term "heroic ideal" is a bit broad.  Let me, however, try to give you some
answers anyway.

Several heroic ideals are mentioned
immediately in Beowulf, particularly in the opening
lines:

So. The
Spear-Danes in days gone by

and the kings who ruled them
had courage and greatness.

We have heard of those princes'
heroic campaigns. (Heaney
translation)

Among the heroic
ideals suggested here and in the ensuing lines are the
following:

• courage
  (2)


• heroism (3)


• skill in
  fighting (3)


• victory in battle
  (9-11)


• generosity
  (20-21)


• concern for his people
  (20-21)


• loyalty to God
  (72)


• loyalty to those who are loyal to him
  (72)


• restraint in the exercise of his power
  (73)


• providing for his loyal followers
  (80-81)


• loyalty to family
  (84-85)

Perhaps the most obvious paradox
concerning heroism is that no matter how good a hero is, his death is inevitable.  Not
even the brave, generous, and loyal can escape mutability and death.  This fact is
implied in line 21 and is openly emphasized in lines 26-52. It is implied again in lines
54-64. The entire poem, of course, will emphasize this paradox: Beowulf, the great hero,
will die just as surely as anyone else, and the poem that begins by describing one
magnificent funeral will end by describing another one as well.