As a general rule of thumb, a run capacitor will far outlast the same motor's start capacitor. A run cap will also fail or wear differently than a start cap, making troubleshooting slightly more involved.
When a run capacitor begins to perform outside the allowable range, it is most often indicated by a dropping of the rated capacitance value (the microfarad value has gone down). For most standard motors, a run capacitor will have a "tolerance" specified describing how close to the rated capacitance value that the actual value may be. This will be usually +/- 5 to 10%. For most motors, as long as the actual value is is within the 10% mark of the rated value, you're in good shape. If it drops outside of this range, you'll need to replace it.
In some cases, due to a defect in a capacitor's construction or sometimes caused by a non-capacitor related motor issue, a run capacitor will bulge from internal pressure. For most modern run capacitor designs, this will open the circuit, disconnecting the internal spiral membrane as a protective measure to prevent the capacitor from popping open.
If its bulging, time to replace. If you measure no continuity across the terminals, it is also time to replace.
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