Chapter 3: Functions of Several Variables
Section 3.2: Limits and Continuity
Show that for f=x sin1/y+y sin1/x the bivariate limit at the origin is zero, but both of the iterated limits limy→0limx→0f and limx→0limy→0f fail to exist.
Hint: Show f≤x+y≤2x2+y2.
To show that L is the bivariate limit at the origin, find δε so that x2+y2<δ⇒fx,y−L<ε.
Consider, then, the following annotated estimate for fx,y−0 = fx,y.
=x sin1/y+y sin1/x
≤x |sin(1/y)|+|y| |sin(1/x)|
Inequalities 4 and 5
Consequently, δ=ε/2, that is, x2+y2<ε/2⇒fx,y<ε.
Figure 3.2.21(a) compares 2x2+y2 with fx,y, the first in green, the second, in red. The green surface lies above the red surface, indicating that near the origin, 2x2+y2 is greater than fx,y.
Figure 3.2.21(a) f in red, 2x2+y2 in green
The iterated limit limy→0limx→0f does not exist: the inner limit fails to exist because of infinite oscillation in sin1/x. Likewise, the iterated limit limx→0limy→0f does not exist: the inner limit fails to exist because of infinite oscillation in sin1/y.
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