In a blender, whirl strawberries, 1/2 cup sugar, and 1 cup water until very smooth. Press blended strawberry mixture through a fine strainer (discard residue from strainer) into a tall pitcher (at least 3 qt.). Whisk in lime juice, 7 cups water, and more sugar to taste. Chill until cold, at least 1 hour or up to 1 day. Serve over ice.
Nutritional analysis per cup.