L'Hospital's Rule

• if or
  
  

• is differentiable in a punctured neighbourhood of ，and
  
  

• (which is allowed to be )
  
  

• then

L'Hospital's Rule

• also allows: or one-sided limits
  
  

• typical cases: etc.

Example

Proof (for the case)

• let ，
  
  

• apply Cauchy's MVT on near
  
  

• 
  
  

Exercise

• find following limits：
  
  

• 
  
  

• 
  
  

Some Counterexamples

• necessity of ,
  e.g.,
  
  

• necessity of the existence of
  e.g., , while does not exist
  
  

• necessity of near
  e.g.,： does not exist, but

Exercise

• find following limits:
  
  

• 
  
  

• 
  
  

• 
  
  

Example

• find
  
  

• apply the rule：
  
  
  

• transform to get：
  

Summary

• L'Hospital's Rule
  
  
• Conditions and Their Necessities
  
  
• Typical Case and How to Handle Them