Limits and Continuity

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Section 15.2

• Limits and Continuity

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LIMIT OF A FUNCTION OF TWO VARIABLES
Let f be a function of two variables whose
domain D includes points arbitrarily close to
(a, b). Then we say the limit of f (x, y) as
(x, y) approaches (a, b) is L and we write if
for every number e gt 0 there is a corresponding d
gt 0 such that
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A LIMIT RESULT
If f (x, y) ? L1 as (x, y) ? (a, b) along a
path C1 and f (x, y) ? L2 as (x, y) ? (a, b)
along a path C2, where L1 ? L2, then
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EXAMPLES
Graph of Example 2.
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CONTINUITY
A function f of two variables is called
continuous at (a, b) if We say f is continuous
on D if f is continuous at every point (a, b)
in D.
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EXAMPLE
is continuous except
The function
on the parabola y x2.
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FUNCTIONS OF THREE VARIABLES
There are analogous definitions for the limit and
continuity for functions of three (or more)
variables.