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Limits of Multivariate Functions

By David A. Smith

In this topic of multivariate calculus we discuss limits of multivariate functions. First we informally define open and closed sets and then give a strict definition of a limit of a function with several variables using epsilons and deltas. We then state several theorems concerning the behavior of limits of multivariate functions. We work through several examples on how to compute limits of multivariate functions by using algebra and other techniques. We also discuss the importance of how to determine whether or not a limit exists. Examples on how to show a limit does exist using epsilons and deltas are also illustrated.  

Definition (Open and Closed Sets) An open disk (or ball) centered at the point is the set of all points such that



for . If the boundary of the disk is included, the disk is closed.

    Open and closed disks are analogous to open and closed intervals on a coordinate line. A point is an interior point of a set   in if some open disk centered at is contained entirely within . If is the empty set, or if every point of is an interior point, then is an open set. The point is a boundary point of if every open disk centered at contains both points that belong to and points that do not. The collection of all boundary points of is called the boundary of , and is a closed disk if it contains its boundary.

Definition (Limit of a Multivariate Function) Let be a real-valued function, then



means that for each given number , there exists a number so that if is a point in the domain of then

Definition (Limit of a Function of Two Variables) Let be a real-valued function, then



means that for each given number , there exists a number so that if is a point in the domain of then

    When considering we need to examine the approach of to from two directions (the left-hand limit and the right-hand limit). However, for functions of two variables, we write to mean that the point is allowed to approach along any curve in the domain of that passes through .
    It is often possible to show that a limit does not exist by showing that the limit has different values depending on which curve is used; however, it is impossible to try to prove that exists by showing the limiting value of is the same along every curve that passes through because there are too many such curves.

Proposition (Limit Rules for Functions of Two Variables) Suppose

,  



and is a real number. Then

        (i) (Scalar Multiple Rule)   
    
        (ii) (Sum Rule)   
    
        (iii) (Product Rule)
    
        (iv) (Quotient Rule)      
    

Example (Limit of a Function of Two Variables) Evaluate, if the limit



    Solution. We find,
    



Example (Limit of a Function of Two Variables) Evaluate the limit of the multivariate function  


    
    Solution. We have along the -axis:



We have along the -axis



We have along the line



Therefore, the limit does not exist.  

Example (Limit of a Function of Two Variables) Evaluate the limit of the multivariate function  



    Solution. We have along the curves :



But along the curve , we have



Therefore the limit does not exist.  

Example (Limit of a Function of Two Variables) Evaluate the limit of the multivariate function  



    Solution. Computing the limit along the curves , and we begin to suspect that the limit exists. Using the definition of a limit of two variables we prove that the limit exists and is 0. Let be given. We want to find such that



that is,  



But since so

  

Thus if we choose and let then




Example (Limit of a Function of Two Variables) Evaluate the limit of the multivariate function  



    Solution. We have












Example (Limit of a Function of Two Variables) Evaluate the limit of the multivariate function  



    Solution. We have, along the path







Therefore, the limit does not exist because as the path changes so does the value

Example (Limit of a Function of Two Variables) Evaluate the limit of the multivariate function  

where is a constant.

    Solution. We have










Example (Limit of a Function of Two Variables) Evaluate the limit of the multivariate function  



    Solution. Along the line we have and along the path we have Therefore, the limit does not exist.  

Example (Limit of a Function of Two Variables) Find a function and a point such that



    Solution. Consider the function and the point Then



for some values of and What does this say about




Example (Limit of a Function of Two Variables) Consider the function Notice that
     


Is it true, then, that



exists?
     
     Solution. No because in fact it does not. Consider along the path We have
     






  
  
which of course varies as the path does. Therefore the limit of the multivariate function does not exist.

Multivariable Calculus Books