Question
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Evaluat

Ask by Coles Deleon. in Ethiopia
Dec 09,2024

Upstudy AI Solution

Tutor-Verified Answer

Answer

The limit is or .

Solution

Calculate the limit .
Evaluate the limit by following steps:
  • step0: Evaluate using L’Hopital’s rule:
  • step1: Use the L’Hopital’s rule:
  • step2: Find the derivative:
  • step3: Use the L’Hopital’s rule:
  • step4: Find the derivative:
  • step5: Simplify the expression:
  • step6: Use the L’Hopital’s rule:
  • step7: Find the derivative:
  • step8: Rewrite the expression:
  • step9: Rewrite the expression:
  • step10: Calculate:

    The limit is equal to or .

Answered by UpStudy AI and reviewed by a Professional Tutor

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Explain
Simplify this solution

Beyond the Answer

To evaluate the limit
we start by recalling the Taylor series expansion of around :
Next, we subtract from :
Now, we substitute this expression into our limit:
As approaches , the terms and go to . Thus, we can take the limit:
Therefore, the answer is

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