Chapter 3: Functions of Several Variables
Section 3.2: Limits and Continuity
Prove that the bivariate limit at the origin for f=x4+y4x2+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.
Add a positive term to numerator
Consequently, δ=ε, that is, x2+y2<ε⇒fx,y<ε.
Figure 3.2.17(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 3.2.17(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)
<< Previous Example Section 3.2
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2021. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
What kind of issue would you like to report? (Optional)