There is no crosswind limit for the approach, it is simply for the take off and landing.
It is very simple to have an FMC page up displaying the calculated crosswind, and therefore monitoring it is not a particular problem.
The reported wind by the tower is the relevant number.
Crosswinds are more often than not subject to gust variation, so that has to be taken into account on a common sense basis.
The crosswind limit is rarely limiting in the structural sense, rather it is a demonstrated safety limit set by the manufacturer.
There are often different limits set for Take off and landing, and those are further modified for dry, wet and slippery runway conditions.