Calculus 2nd Edition Rogawski

SECTION 15.2 Limits and Continuity in Several Variables 795
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EXAMPLE 2 Evaluating Limits by Substitution Show that
y
f (x, y) = 3x + y
x 2 + y 2 + 1
is continuous (Figure 4). Then evaluate lim f (x, y).
(x,y)→(1,2)
Solution The function f (x, y) is continuous at all points (a, b) because it is a rational
function whose denominator Q(x, y) = x 2 + y 2 + 1 is never zero. Therefore, we can
x
evaluate the limit by substitution:
3x + y
lim
(x,y)→(1,2)
FIGURE 4 Top view of the graph
x 2 + y 2 + 1 = 3(1) + 2
1 2 + 2 2 + 1 = 5 6
f (x, y) =
3x + y
x 2 + y 2 + 1 . If f (x, y) is a product f (x, y) = h(x)g(y), where h(x) and g(y) are continuous,
then the limit is a product of limits by the Product Law:
( ) ( )
lim f (x, y) = lim h(x)g(y) = lim h(x) lim g(y)
(x,y)→(a,b) (x,y)→(a,b) x→a y→b
EXAMPLE 3 Product Functions Evaluate lim
(x,y)→(3,0)
x3
sin y
y .
Solution The limit is equal to a product of limits:
lim
(x,y)→(3,0)
x3
sin y
y
=
(
lim
x→3 x3 )(
lim
y→0
)
sin y
= (3 3 )(1) = 27
y
Composition is another important way to build functions. If f (x, y) is a function
of two variables and G(u) a function of one variable, then the composite G ◦ f is the
function G(f (x, y)). According to the next theorem, a composite of continuous functions
is again continuous.
THEOREM 2 A Composite of Continuous Functions Is Continuous If f (x, y) is continuous
at (a, b) and G(u) is continuous at c = f (a, b), then the composite function
G(f (x, y)) is continuous at (a, b).
EXAMPLE 4 Write H (x, y) = e −x2 +2y as a composite function and evaluate
lim H (x, y)
(x,y)→(1,2)
Solution We have H (x, y) = G ◦ f , where G(u) = e u and f (x, y) = −x 2 + 2y. Both
f and G are continuous, so H is also continuous and
lim H (x, y) = lim
(x,y)→(1,2) (x,y)→(1,2) e−x2 +2y = e −(1)2 +2(2) = e 3
We know that if a limit
lim
(x,y)→(a,b)
f (x, y) exists and equals L, then f (x, y) tends to
L as (x, y) approaches (a, b) along any path. In the next example, we prove that a limit
does not exist by showing that f (x, y) approaches different limits along lines through the
origin.