Chapter 3: Functions of Several Variables
Section 3.2: Limits and Continuity
Extend f=x4+x2+x y2+y2x2−x2y+y2 to a function gx,y that is continuous at the origin.
Maple determines the bivariate limit at the origin to be 1.
Context Panel: Assign Function
fx,y=x4+x2+x y2+y2x2−x2y+y2→assign as functionf
Context Panel: Evaluate and Display Inline
Context Panel: Limit (Bivariate)
fx,y = x4+x⁢y2+x2+y2−x2⁢y+x2+y2→bivariate limit1
To see why the bivariate limit at the origin might have the value 1, divide the numerator and denominator of f to put fx,y into the form
The rational functions in the numerator and denominator of this form of f both tend to zero as x,y→0,0. Indeed, Maple provides the following bivariate limits, obtained either through the Context Panel, or by application of the limit command with the appropriate syntax.
x4+x y2x2+y2→bivariate limit0
limitx4+x y2x2+y2,x=0,y=0 = 0
limitx2yx2+y2,x=0,y=0 = 0
Auxiliary Limit (1)
Auxiliary Limit (2)
Clearly, then, the bivariate limit of f as x,y→0,0 will necessarily be 1, in which case the required extension is
Analytic justification for Auxiliary Limit (1) is based on the following estimate.
Inequality 3, Table 3.2.1
Inequalities 4, 6, and 7, Table 3.2.1
Consequently, x4+x y2x2+y2≤x2+y2+x2+y2, and the limiting value of zero is established.
Analytic justification for Auxiliary Limit (2) is based on the following estimate.
Inequalities 5 and 6, Table 3.2.1
Consequently, x2yx2+y2≤x2+y2, and the limiting value of zero is established.
Indeed, limitfx,y,x=0,y=0 = 1.
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