Combine first 4 ingredients in a small saucepan; bring to a boil. Cover and remove from heat; let stand 15 minutes. Strain cranberry mixture through a fine mesh sieve into an 11 x 7-inch baking dish; discard solids. Cool to room temperature; stir in lime juice. Cover and freeze for about 45 minutes. Stir cranberry mixture every 45 minutes until completely frozen (about 3 hours). Remove mixture from freezer; scrape entire mixture with a fork until fluffy.
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