How many wives did Charlemagne have?
Four, but he also had quite a number of concubines...
Officially, he had four:
Desiderata (married 770, annulled in 771)
Hildegard of the Vinzgau (married 771, died 783)
Fastrada (married 784, died 794)
Luitgard (married 794)
But he had concubines and other relationships. All told, he had 18 children in total, with eight of his ten known wives and concubines. But apparently his kids had a thing for illegitimate children because Charlemagne only had four legitimate grandchildren.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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