To make the simple syrup, combine 1 cup water with 1 cup granulated sugar in a small saucepan. Heat over medium until the sugar is dissolved and the mixture starts to boil. Remove and cool slightly.
Place the 4 cups of sliced strawberries in a mixing bowl with a lid or one that can be covered with plastic wrap. Combine 1/2 cup of the simple syrup with 1 cup of the cold water and pour over the strawberries. Cover and refrigerate for at least 4 hours or up to 8.
Pour the strawberries and all accumulated juice into a blender and process until smooth.
Set a fine mesh strainer over a pitcher. Pour the strawberry mixture through the strainer to remove the seeds. Do this process twice if needed. Use a spoon to press the juice and pulp through the mesh as needed. Discard the seeds and pulp.
Add 7 cups of cold water and the fresh lime juice to the strawberry mixture in the pitcher. Refrigerate until ready to serve or serve immediately over ice garnished with sliced lime.
Pass the remaining 1/2 cup of simple syrup to your guests if they prefer a sweeter drink.