To show that is the bivariate limit at the origin, find so that .
Consider, then, the following annotated estimate for = .
|
|
|
|
|
Inequality 3
Table 3.2.1
|
|
|
|
|
|
Inequalities 4 and 5
Table 3.2.1
|
|
|
|
|
|
Consequently, , that is, .
Figure 3.2.21(a) compares with , the first in green, the second, in red. The green surface lies above the red surface, indicating that near the origin, 2 is greater than .
|
Figure 3.2.21(a) in red, 2 in green
|
|
|
The iterated limit does not exist: the inner limit fails to exist because of infinite oscillation in . Likewise, the iterated limit does not exist: the inner limit fails to exist because of infinite oscillation in .