Limits and Continuity of Vector Functions

     The idea is now to extend the notion of a limit of a function and continuity of a function to vector functions by capitalizing on the linearity properties of vector functions.

Definition (Limit of Vector-Valued Functions) Suppose the components function , , and of the vector-valued function



all have finite limits as , where is any real number or . Then the limit of as is the vector



Notice that if the limit of a vector function exists then it is a vector, (and not a scalar function).

Proposition (Limits of Vector-Valued Functions) If the vector-valued functions and are functions of a real variable and is a real-valued function such that all three functions have finite limits as , then











These limit formulas are also valid when provided all limits are finite.

Example (Limits of Vector-Valued Functions) Given and find

    Solution. We have,     





Example (Limits of Vector-Valued Functions) Given and   find  

    Solution. We have,





  

Definition (Continuity of Vector-Valued Functions) A vector-valued function is continuous at means is in the domain of and . Further, a vector function is continuous on an interval if it is continuous at every point in the interval.

Example (Continuity of Vector-Valued Functions) Determine where the vector-valued function



is continuous.

    Solution. The function is continuous for all real numbers in its domain which is
    

    
Example (Continuity of Vector-Valued Functions) Determine where the vector-valued function   



where is continuous.  

    Solution. The function is continuous for all real numbers in its domain which is because



and









for all real numbers such that

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Limits And Continuity Of Vector Functions
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/limits-and-continuity-of-vector-functions.html