Chapter 3: Functions of Several Variables
Section 3.2: Limits and Continuity
Prove that the bivariate limit at the origin for f=2 x3−y3x2+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.
≤2 x3+2 y3x2+y2
Add y3 to numerator
=2 x x2+2 y y2x2+y2
≤2 x2 x2+y2+2 y2 x2+y2x2+y2
Inequalities 4 and 5
Consequently, δ=ε/2, that is, x2+y2<ε/2⇒fx,y<ε.
Figure 3.2.14(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 3.2.14(a) f in red, 2x2+y2 in green
Maple corroborates these results by computing the bivariate limit, here accessed through the Context Panel.
Context Panel: Limit (Bivariate)
2 x3−y3x2+y2→bivariate limit0
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