Multivariable Epsilon-Delta Limit Definitions
Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
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 
.
Contributed by: Spencer Liang (The Harker School) (March 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
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.
Permanent Citation
Nonexistence of a Multivariable Limit
Laura R. Lynch
Epsilon-Delta Definition of Limit
Ferenc Beleznay
Solids of Revolution
Abby Brown and MathematiClub (Torrey Pines High School)
Solids of Known Cross Section
Abby Brown (Torrey Pines High School)
Double Integral for Volume
Anton Antonov
A Case of the Epsilon-Delta Definition of a Limit
Joseph F. Kolacinski
Sine Wave Example of the Epsilon-Delta Definition of Limit
Geoffrey F. Miller, Daniel C. Cheshire, Nell H. Wackwitz, Joshua B. Fagan
Volumes of Revolution Using Cylindrical Shells
Stephen Wilkerson (Towson University)
Curvas de nivel (Spanish)
Jorge Gamaliel Frade Chávez
Cone, Tent, and Cylinder
George Beck
