Chapter 3: Functions of Several Variables
Section 3.2: Limits and Continuity
If fx,y=2 x2−6 x y+5 y2, prove that x2+y2<ε/8=δ ⇒ |fx,y|<ε.
The requisite estimates are shown below.
2 x2−6 x y+5 y2
≤2 x2+6x y+|5 y2|
=2 x2+5 y2+6x y
If 8x2+y2 is to be less than ε, and if δ=x2+y2 is the radius of a circular neighborhood about the origin, then 8 δ2<ε⇒δ=ε/8.
Figure 3.2.19(a) compares 8x2+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, 8x2+y2 is greater than fx,y.
Figure 3.2.19(a) f in red, 8x2+y2 in green
<< 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)