In a small saucepan, combine the water with the sugar and cook over moderate heat, stirring, just until the sugar has dissolved. Let the sugar syrup cool. Add the sugar syrup to the strawberry puree along with the lime juice. Transfer the granita mixture to a large, shallow nonreactive roasting pan and freeze just until ice crystals begin to form around the edge of the pan, about 30 minutes. Using a fork, scrape the ice crystals into the center of the roasting pan. Continue to freeze the granita for about 2 hours, scraping the ice crystals into the center every 30 minutes, until the granita is fluffy and icy. Transfer the strawberry granita to bowls and serve.