Place the water, sliced ginger root and sugar in a pot and bring to a boil.
Reduce heat to low/med and simmer for 5 minutes, then set aside and allow the tea to cool to room temperature. Keep in mind that if the ginger tea is too hot when you add the ginger bug, it will kill the beneficial yeast and bacteria needed for fermentation.
Place a fine mesh strainer over a bowl and pour mixture into it to separate the ginger from the liquid.
Add the ginger bug and lime/lemon juice (if using) to the liquid and mix with a wooden spoon.
Pour the liquid into flip-top bottles (leaving one inch of head space) and ferment for 3-6 days. During the fermentation process, "burp" the bottles by opening them every 1-2 days so the carbonation doesn't build up too much. Some people have reported that their bottles exploded because they let them ferment for way too long without burping them. I've been making fermented drinks for years and have never had that happen, but I have forgotten about them for a few days and ended up with an extra fizzy bottle that poured out like champagne. I open my bottles on our deck now just in case. :)
The ginger ale is ready when it's fizzy and not overly sweet. If you check it after a week and it's still too sweet, you can let it ferment for a bit longer so that the good guys can consume more of the sugar and turn them into probiotics.