To show vitamin deficiencies they have to test specifically for them. Potassium can be tested but is sometimes coupled with magnesium deficiency and magnesium can't be tested well.
Some deficiencies might show up in standard blood work, like both b12 and iron can make the red count go down. Red count is part of basic blood work. If it is low, one can suspect a deficiency and investigate further. That said, those deficiencies might not always alter red count to a big degree so it might be missed.
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