From Nookipedia, the Animal Crossing wiki.
The Giant Petaltail is a dragonfly found in August, September, and October. It is considered a rare insect, selling for 8,000 bells. It is dark in color, and is easily identifiable due to its immense size and evasive flying ability, making it difficult to catch. It also can fly over cliffs, unlike other dragonflies. It has the largest picture of all bugs in the encyclopedia.
 Donating to the Museum
 In City Folk
"Hootie hoo! It's like looking at a living ancestor of today's dragonflies! A vile ancestor... It's so big that when I tell people I spotted one, no one wants to believe me. Folks claim I'm being daft. It's understandable, wot? When I look at this thing, I think I might very well BE hallucinating..."
The giant petaltail can be found on the third tier of the insect section of the museum, at the top of the ramp, with the lantern fly, fruit beetle, and scarab beetle, to name a few. During the evening hours, it can be found sitting on the sign in that section of the museum.
 Bug Encyclopedia Information
 City Folk
 Further Information