In detection of CO₂, lime water turns milky due to formation of

Limewater absorbs carbon dioxide, resulting in a milky solution. This is caused by the formation of an insoluble suspension of calcium carbonate.

$\underset{\text{ lime water }}{\mathrm{Ca}(\mathrm{OH}{)}_2}+{\mathrm{CO}}_2⟶\underset{\text{ milkiness }}{{\mathrm{CaCO}}_3}+{\mathrm{H}}_2\mathrm{O}$