When carbon dioxide is passed through limewater, you see a milky white precipitate, #CaCO_3 (s)#. Lime water is saturated #Ca(OH)_2(aq)#; #Ca(OH)_2# is sparingly soluble in water. #CaCO_3#, alike with most carbonates, is pretty insoluble stuff; much more so that #Ca(OH)_2#.
Its reaction with #CO_2 (g)# may be represented:
#Ca(OH)_2(aq) + CO_2(g) rarr CaCO_3(s)darr + H_2O(l)#
Note that this equation is balanced.
If you keep bubbling #CO_2#, the white solid should go up. You should form the bicarbonate, #Ca(HCO_3)_2#, which is soluble in water, and the reaction is outside the A level syllabus. As a further tip, the best source of carbon dioxide outside of a cylinder, is carbonated water (fizzy water; water supersaturated with #CO_2(g)#); you can cheaply pick up litres of this in a supermarket.
None of what I have written here is a substitute for doing the reaction in a laboratory
