Preparation Bring sugar, ginger, and 1/3 cup water to a boil in a small saucepan, stirring until sugar dissolves. Remove from heat; let steep for 10 minutes. Strain through a fine-mesh sieve into a medium pitcher (you should have 1/2 cup); discard ginger. Cover and chill syrup. DO AHEAD: Can be made 1 week ahead. Keep chilled. Fill four 6-ounce glasses with ice cubes. Add carrot juice and lime juice to ginger syrup in pitcher; stir well. Divide elixir among prepared glasses.
Preparation Bring sugar, ginger, and 1/3 cup water to a boil in a small saucepan, stirring until sugar dissolves. Remove from heat; let steep for 10 minutes. Strain through a fine-mesh sieve into a medium pitcher (you should have 1/2 cup); discard ginger. Cover and chill syrup. DO AHEAD: Can be made 1 week ahead. Keep chilled. Fill four 6-ounce glasses with ice cubes. Add carrot juice and lime juice to ginger syrup in pitcher; stir well. Divide elixir among prepared glasses.