Chapter 3: Functions of Several Variables
Section 3.2: Limits and Continuity
Example 3.2.10
For f=x4y4x2+y43, show that the bivariate limit at the origin does not exist.
Solution
Mathematical Solution
Along either axis, the function has the value zero, that is, fx,0=f0,y=0. Hence, the limits approaching the origin along either axis will be zero.
Along y=x, fx,x=1/8, so the limit along this curve will be 1/8.
Since the limit depends on the direction of approach to the origin, the bivariate limit at the origin does not exist.
Maple Solution - Interactive
Define the function fx,y
Context Panel: Assign Function
fx,y=x4y4x2+y43→assign as functionf
Evaluate f along the axes
Context Panel: Evaluate and Display Inline
fx,0 = 0
f0,y = 0
Evaluate f along y=x3/2
fx,x = 18
Along either axis, the limit of f will be zero; along the curve y=x, it will be 1/8. Since the limit depends on the direction of approach to the origin, the bivariate limit at the origin does not exist.
Alternatively, access Maple's bivariate limit through the Context Panel.
Context Panel: Limit (Bivariate) (Fill in the Limit Point dialog as per Figure 3.2.10(a).)
Figure 3.2.10(a) Limit Point dialog
fx,y = x4y4y4+x23→bivariate limit0..427
The return of a range indicates that the limit does not exist.
Maple Solution - Coded
Compute the bivariate limit with Maple
Define the function f.
f≔x,y→x4y4x2+y43:
Apply the bivariate limit command.
limitfx,y,x=0,y=0
0..427
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