When you have some time, give this a try:

Daytime Lime Sorbet (Sugar-Free)

2 cups Splenda or Xylitol, or other sugar substitute
1 cup water
1 & 1/2 cups lime juice
1 tsp grated lime zest (if available)

Combine water and sugar substitute in a saucepan.
Bring to simmer.
Cook, stirring, about 3 minutes, until the sugar sub dissolves. Cool.
Combine 1 & 1/2 cups of the syrup with the lime juice.
Taste, and add more syrup, as needed, to obtain desired sweetness. *
Stir in the lime zest.
Chill in refrigerator 2 - 3 hours.
When chilled, transfer to freeze in ice cream freezer.
Or put in fridge freezer, removing every half hour, and beating it with an electric beater, until frozen and creamy.
The more times you remove and beat it, the creamier it will be.

* IMPORTANT: Make it sweeter than you think it should be, because freezing lessens the sweet taste. Use any leftover sweet syrup for coffee, grapefruit, etc.