Chapter 3: Functions of Several Variables
Section 3.2: Limits and Continuity
Example 3.2.3
For f=x−yx2+y2, 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→0x−m xx2+m2x2
=limx→01−mx1+m2
=1−m1+m2 limx→01x
which does not exist because 1/x 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−yx2+y2→assign as functionf
Evaluate limx→0fx,m x
Calculus palette: Limit operator
Context Panel: Simplify≻Assuming Real Range (Complete dialog as per figure below.)
limx→0fx,m x→assuming real rangeundefined
Since the limit fails to exist along any line that passes through the origin with positive slope, 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 3.2.3(a).)
Figure 3.2.3(a) Limit Point dialog
fx,y = x−yx2+y2→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−yx2+y2:
Apply the limit command to fx,m x, and make the assumption that m>1. Press the Enter key.
limitfx,m x,x=0 assuming m>1
undefined
For corroboration, apply Maple's bivariate limit command
Obtain the bivariate limit.
limitfx,y,x=0,y=0 = undefined
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