Both a limit gauge and a coordinate measuring machine can decide whether a cap is acceptable, and shops often debate which is better. The question is misstated: they answer different questions, and a competent inspection plan uses both for different purposes.
The gauge is fast, requires no interpretation, and measures the functional combination that matters: will this cap mate with any holder made to the standard. It is the right tool for lot acceptance on a line, where the decision has to be made in seconds and the operator should not need a metrology background.
The CMM gives numbers, with uncertainty, on individual features. It is the right tool for first article inspection, for diagnosing a gauge failure, for tool wear trend analysis, and for disputes. It is slow, it needs a programme and a competent operator, and it measures features rather than function.
A sound plan gauges every lot and measures a sample periodically, so that a drift in the process is seen as a trend before it becomes a rejection. When a lot fails gauging, the CMM says whether the cause is a single outlier or a shifted mean, and that determines whether the response is sorting or a tool change.
