Welcome to the Orchid Board! From your description, your plant is a Phalaenopsis, sometimes commonly called a moth orchid. As others have mentioned, when the flowers fall off, that "limb" as you are calling it, will most likely start to turn brown. Many of us do not cut it off until is is completely brown, because while it is still green, it is possible for it to start a secondary spike and bloom again. Also, it's possible that the plant derives some nutrition from a green spike.
In most cases, a Phalaenopsis like this will produce a brand new spike from a different point on the plant approximately once a year. So, yes, you will be many months between blooming cycles, but with proper care, it will bloom again reliably.
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Cheri
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