How does carbon dioxide react with limewater \[?\]

Hint: First we have to know the limewater is a dilute aqueous solution of calcium hydroxide (\[Ca{\left( {OH} \right)_2}\]). Carbon dioxide is an acidic oxide, so it reacts with sodium hydroxide to form salt and water. The solution turns cloudy, when carbon dioxide gas bubbles through a solution of calcium hydroxide.

Complete answer:
Carbon dioxide reacts with lime water to form calcium carbonate, which precipitates out of the solution.
The reaction between lime water and carbon dioxide will result in the formation of an insoluble solid called calcium carbonate (\[CaC{O_3}\]). Its chemical reaction is written as follows
\[Ca{\left( {OH} \right)_2}\left( {aq} \right) + C{O_2}\left( g \right) \to CaC{O_3}(s) + {H_2}O\left( l \right)\]
The obtained solution will turn milky. This reaction is actually used to test for the presence of carbon dioxide.
If we continue to bubble the carbon dioxide through the limewater another acid - base reaction occurs which results in the precipitate dissolving to give soluble calcium hydrogen carbonate. Its chemical reaction is written as follows
\[CaC{O_3}(s) + {H_2}O\left( l \right) + C{O_2}\left( g \right) \to Ca{\left( {HC{O_3}} \right)_2}\left( {aq} \right)\]
When this solution evaporates the reverse reaction occurs resulting in the formation of stalactites and stalagmites.

Note:
Note that calcium carbonate is insoluble in water and forms a precipitate which would turn limewater milky. Carbon dioxide is used as a refrigerant, in fire extinguishers, blasting coal, foaming rubber and plastics, etc. Since carbon is in carbon dioxide, which is a greenhouse gas that helps earth hold the energy received from the sun so it doesn't all escape back into space.