Chapter 3: Functions of Several Variables
Section 3.2: Limits and Continuity
Prove that the bivariate limit at the origin for f=x y x2−y2x2+y2 is zero.
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 y x2−y2x2+y2
≤x y x2+y2x2+y2
≤ x2+y2 x2+y2
Inequalities 4 and 5
Consequently, δ=ε, that is, x2+y2<ε⇒fx,y<ε.
Figure 3.2.15(a) compares x2+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, x2+y2 is greater than fx,y.
Figure 3.2.15(a) f in red, x2+y2 in green
Maple corroborates these results by computing the bivariate limit, here accessed through the Context Panel.
Context Panel: Limit (Bivariate)
x y x2−y2x2+y2→bivariate limit0
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