Squeeze Rule

    This topic states the squeeze rule and through examples shows how to use it. Basically, the idea is to bound a function on both sides by functions whose limits can be more easily computed; and thus in the process squeeze the value of the limit of the original function out. The two special trigonometric limits can be proven using the squeeze rule.

Proposition (Squeeze Rule) Let and be functions. If

    (i) on an open interval containing and

    (ii)  
    
then

Example (Squeeze Rule) Use the squeeze rule to find the limit of at

    Solution. Let's plot a graph to get an idea.



However we are interested in this function around so a better graph is



Knowing that the cosine function is always less than or equal to one, we see that when we have So we can bound this function by the lines and as shown:



Now we have on the open interval containing   Since we have by the squeeze rule.

Example (Squeeze Rule) Use the squeeze rule to find the limit of at

    Solution. Let's plot a graph to get an idea.
    



However we are interested in this function around so a better graph is




Knowing that the sine function is always less than or equal to one, we see that when we have



So we can bound this function by the curves and as shown:



and up close on the interval we have:



Since we have by the squeeze rule.

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Squeeze Rule
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/squeeze-rule.html