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Exercise 80: Calculate the limit  .

Solution: We use the theorems

Theorem: If  for a sequence we have , where q < 1, then .

Theorem: If  for a sequence we have , where q >1 1, then

We get

Using the definition of  n! = n * (n-1) * (n-2) * ... * 2 * 1, so n! = n * (n-1)!, we get

 

Using the formula  , we get

 

Thus , Using the theorem, we get