Answer in brief.
Why does lime water turn milky when it comes in contact with air?
When air is blown into limewater, the solution turns milky. This confirms the presence of carbon dioxide CO2 in the air. This is because CO2 reacts with limewater Ca(OH)2 and forms insoluble calcium carbonate CaCO3 which is 'white' in color.
The suspended particles of white calcium carbonate settle down after some time.
The chemical reaction involved is:
Ca(OH)2(aq)+CO2(g)→CaCO3(s)+H2O(l)