Chapter 3: Functions of Several Variables
Section 3.2: Limits and Continuity
Prove that the bivariate limit at the origin for f=2 x5+2 y32 x2−y2x2+y22 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 x5+2 y32 x2−y2x2+y22
=2 x5−y5+2 x2y3x2+y22
≤2 x5+y5+2 x2y3x2+y22
≤2x x4+y y4+2 x2 y y2x2+y22
Inequalities 4 and 5
Consequently, δ=ε/2, that is, x2+y2<ε/2⇒fx,y<ε.
Figure 3.2.18(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 3.2.18(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 x5+2 y3 2 x2−y2x2+y22→bivariate limit0
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