What in AU's is the star Alpha Centauri's habitable zone?

Answer 1

The [related links] contain all the information you want. Habitable zones for each of the major stars in AUs.
You could perhaps try calculating the limits of the Habitable zones for each of the stars in the Alpha Centauri system yourself, using the 'Habitable Zone Calculator,' which is based on findings from an up-to-date research paper.
Using the calculator and taking temperature/luminosity values given in Wikipedia, these were the limits I got for Alpha Centauri A, B and Proxima for the minimum inner edge of the habitable zone (Runaway Greenhouse limit, 1 Earth mass, 1 atmosphere):

1.2 AU, 0.71 AU, 0.044 AU
Again using the calculator, the limits I got for alpha Centauri A, B and Proxima for the maximum outer edge of the habitable zone (1 Earth mass, 8 carbon dioxide atmospheres) were:
2.1 AU, 1.2 AU, 0.09 AU
A more conservative measure of the maximum outer edge of the habitable zone, the '1st Condensation Limit', the distance from the star where the average temperature of the planet is cold enough for the carbon dioxide gas in the atmosphere to freeze out, is not a scenario included in the calculator. This is because the paper the calculator is based on assumes that after carbon dioxide freezes out and the planet glaciates there would be an inevitable build up of carbon dioxide; rocks deep under an increasingly glaciated planet would no longer be reacting with the atmosphere and so would not be removing carbon dioxide which is being pumped out all the time into the atmosphere from volcanoes, resulting in a warming feedback. However, a rough estimate of the 1st Condensation Limit for a 1 atmosphere planet around Alpha Centauri A, B and Proxima would be:
1.65 AU, 1.0 AU, 0.07 AU
It's a shame the 'Habitable Zone Calculator' is so buggy and that the site that hosts the calculator doesn't offer an Excel version to download and work from.