Chapter 3: Functions of Several Variables
Section 3.2: Limits and Continuity
Extend f=tan⁡x⁢ytan⁡x⁢tan⁡y 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 1. Unfortunately, showing f−1→0 by the "usual" techniques of estimation turns out to be a significant challenge. Instead, a solution based on the algorithm Maple uses for computing bivariate limits is presented. In this algorithm the maxima of f on circles of radius r are computed, and then if it can be shown that these maxima tend to L as r→0, it will have been established that the bivariate limit of f at the origin is L. To this end, express f in polar coordinates and find the extrema for fixed r as a function of θ.
Define the function fx,y
Context Panel: Assign Function
fx,y=tan⁡x⁢ytan⁡x⁢tan⁡y→assign as functionf
Change to polar coordinates
F≔fr cosθ,r sinθ = tan⁡r2⁢sin⁡θ⁢cos⁡θtan⁡r⁢cos⁡θ⁢tan⁡r⁢sin⁡θ
Find the extrema of F on circles of fixed radius r
Simplify the derivative dFdθ.
Solve the equation dFdθ=0.
Simplify the solutions subject to the restriction that r∈0,1.
simplifysolvesimplifyⅆⅆ θ F=0,θ assuming r>0,r<1
Evaluate F at each solution of dFdθ=0, thus obtaining the extreme values of F on the circle of radius r
Obtain, as r→0, the limiting value of the common extreme value of F on the circle of radius r
Context Panel: Limit operator
Expression palette: Evaluation template
limr→0Fx=a|f(x)θ=π/4 = 1
Since the bivariate limit of f at the origin is 1, the required extension is
Indeed, limitfx,y,x=0,y=0 = 1.
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