Chapter 3: Functions of Several Variables
Section 3.2: Limits and Continuity
Example 3.2.7
For f=x yx4+y4, show that the bivariate limit at the origin does not exist.
Solution
Mathematical Solution
Evaluate the limit along the lines y=m x, that is, evaluate
limx→0fx,m x
=limx→0m x2x4+m4x4
=limx→0mx21+m4
=m1+m4 limx→01x2
which does not exist because 1/x2 becomes unbounded at the origin. Therefore, the bivariate limit at the origin does not exist because f becomes unbounded at the origin.
Maple Solution - Interactive
Define the function fx,y
Context Panel: Assign Function
fx,y=x yx4+y4→assign as functionf
Evaluate limx→0fx,m x
Calculus palette: Limit operator
Context Panel: Simplify≻Assuming Positive
limx→0fx,m x→assuming positive∞
Under the assumption that m>0, the limit along any line through the origin does not exist because fx,m x becomes unbounded. Hence, the bivariate limit at the origin does not exist.
Alternatively, access Maple's bivariate limit through the Context Panel.
Context Panel: Evaluate and Display Inline
Context Panel: Limit (Bivariate) (Fill in the Limit Point dialog as per Figure 3.2.7(a).)
Figure 3.2.7(a) Limit Point dialog
fx,y = xyx4+y4→bivariate limitundefined
Maple's declaration that the limit is undefined is equivalent to the more prevalent statement that the limit does not exist.
Maple Solution - Coded
Define the function fx,y.
f≔x,y→x yx4+y4:
Apply the limit command to fx,m x, and make the assumption that m>0. Press the Enter key.
limitfx,m x,x=0 assuming m>0
∞
For corroboration, apply Maple's bivariate limit command
Obtain the bivariate limit.
limitfx,y,x=0,y=0 = undefined
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