Limits of Intermediate Forms

We say that the following limit



has the intermediate form because and as We will consider the following indeterminate forms,



Our main investigative tool will be L'Hospital's rule which says that the limit of a quotient of functions and is equal to the limit of the quotient of their derivatives and , provided that the following conditions are satisfied. It is especially important to verify the conditions regarding the limits of and before applying L'Hospital's rule. Also, keep in mind that L'Hospital's' rule also holds if " " is replaced by

Proposition (L'Hospital's Rule) Suppose and are differentiable functions and on an open interval that contains (except possibly at ). Suppose produces an intermediate form or and that then

Example (Using L'Hospital's Rule Incorrectly) Try to evaluate the limit using L'Hospital's rule.

    Solution. This limit has indeterminate form since
    
  

and  



If  we try to apply L'Hospitals's Rule we find,  
    


but the limit does not exist because of osculating behavior, so we can not use L'Hospital's rule. To correctly find this limit we divide by to find

Example (L'Hospital's Rule with Intermediate Form ) Evaluate the limit using L'Hospital's rule.

    Solution. The limit has the indeterminate form of since and We try to apply L'Hospital's rule to find,
    

Example (L'Hospital's Rule with Intermediate Form )  Evaluate the limit using L'Hospital's rule.

    Solution. The limit has the indeterminate form since and We try to use L'Hospital's rule to find,
    


Example (L'Hospital's Rule with Intermediate Form )  Evaluate the limit using L'Hospital's rule.

    Solution. The limit has the indeterminate form since and We try to use L'Hospital's rule to find,
    


Thus illustrating that L'Hopital's rule can be used multiple times.

Example (L'Hospital's Rule with Intermediate Form ) Evaluate the limit using L'Hospital's rule.

    Solution. The limit has the indeterminate form since and Suppose the limit exists, say Then,







Since and this limit has the indeterminate form of and so we try L'Hospital's rule,  








    

Example (L'Hospital's Rule with Intermediate Form ) Evaluate the limit using L'Hospital's rule.

    Solution. The limit has the indeterminate form since and Suppose the limit exists, say Then,










    

Example (L'Hospital's Rule with Intermediate Form ) Evaluate the limit using L'Hospital's rule.

    Solution. The limit has the indeterminate form since and Suppose the limit exists, say Then,









The limit has the indeterminate form since and So we try to use L'Hospital's rule to find,

  

Example (L'Hospital's with Rule Intermediate Form ) Evaluate the limit using L'Hospital's rule.

    Solution. The limit has the indeterminate form since and We try L'Hospital's rule to find,

  

Example (L'Hospital's Rule with Intermediate Form ) Evaluate the limit using L'Hospital's rule.

    Solution. Let's make a change of variable to simply the express, namely Since as we have







The limit has an indeterminate form and so we try L'Hopital's rule,

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Cite this as:
Limits Of Indeterminate Forms
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/limits-of-indeterminate-forms.html