I knew of the Elves since childhood, but I wasn't present for the reforging.

[It's somewhat surreal for Aragorn to be talking to Frodo like this. Literally just the other day the hobbit was a fully grown adult, and now he's just a little boy.]

Yes. [His smile fades once the child recognized the name of his ancestor.] Elendil and his sons were the ones who battled the Dark Lord of Mordor.

[He refuses to mutter Sauron's name to the youth.]

Narsil was the blade that Elendil wielded, but it wasn't him who slew the Dark Lord.