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
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