Multivariable Epsilon-Delta Limit Definitions

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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
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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.
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