Multivariable Epsilon-Delta Limit Definitions
|
 |
The definition of a limit: The expression  is an abbreviation for: the value of the single-variable function  approaches   as  approaches the value  . More formally, this means that  can be made arbitrarily close to  by making  sufficiently close to  , or in precise mathematical terms, for each real  , there exists a  such that  . In other words, the inequalities state that for all  except  within  of  ,  is within  of  . This definition extends to multivariable functions as distances are measured with the Euclidean metric. In the figure, the horizontal planes  represent the bounds on  and the cylinder is  . No matter what  is given, a  is found (represented by the changing radius of the cylinder) so that all points on the surface  inside the cylinder are between the two planes. |
|
|
|
|
|
|
|
 For the limit of a multivariable function, consider the two-variable function  . (Note that the following extends to functions of more than just two variables, but for the sake of simplicity, two-variable functions are discussed.) The same limit definition applies here as in the one-variable case, but because the domain of the function is now defined by two variables, distance is measured as  , all pairs  within  of  are considered, and  should be within  of  for all such pairs  . As an example, here is a proof that the limit of  is 10 as  . Claim: for a given  , choosing  satisfies the appropriate conditions for the definition of a limit:  (the given condition) reduces to  , which implies that  and  . Now,  by the triangle inequality, and  . If  ,  , and if  ,  . Thus by the choice of  ,  , and because  is arbitrary, an appropriate  can be found for any value of  ; hence the limit is 10.
|
|
|
