Chapter 3: Functions of Several Variables
Section 3.2: Limits and Continuity
Extend f=x2⁢y⁢cos⁡x⁢yx2+y2 to a function gx,y that is continuous at the origin.
The requisite extension assigns to the origin the value of the bivariate limit of f at the origin. Hence, what is required is to show that this limit is zero, a computation summarized below.
Since f−0≤x2+y2, it is clear that f−0→0 as x,y→0,0. Hence, the required extension is
Indeed, limitx2⁢y⁢cos⁡x⁢yx2+y2,x=0,y=0 = 0.
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