Limit Rules

    The basic rules that are used to compute limits are stated. Also included are examples on how to use them and limit rules for the trigonometric functions. One of the most important limit rules is that the limit of a continuous function, such as a polynomial or rational function, is the value of the function at the fixed given value.  
    The following rules can be proven using the formal definition of the limit of a function.

Proposition (Limit Rules) For any real number suppose the functions and both have finite limits at Then

    (i) (Constant) for any constant
    
    (ii) (Limit of )
    
    (iii) (Multiple)
    
    (iv) (Sum)
    
    (v) (Difference)
    
    (vi) (Product)
    
    (vii) (Quotient)
    
    (viii) (Power)   where is a rational number and whenever the limit on the right exists
    
    (ix) (Polynomial) for any polynomial
    
    (x) (Rational) for any rational function where is in the domain of
    

Example (Limit Rules) Find the limit of at

    Solution. By using several limit rules, we have















This is the same as evaluating the rational function at because 3 is in the domain of

Proposition (Trigonometric Functions) If is any real number in the domain of the given function, then

Example (Trigonometric Functions)  Compute the limit of at

    Solution. We have

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Limit Rules
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/limit-rules.html