Chapter 3: Functions of Several Variables
Section 3.2: Limits and Continuity
Extend f=x⁢sin⁡x⁢y2−cos⁡x−cos⁡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 0. Unfortunately, showing f−0→0 by the "usual" techniques of estimation turns out to be a significant challenge. So also is the approach taken in Example 3.2.25 where the maxima of f on circles of radius r are computed, and then shown to tend to L as r→0. At best, a graph of f in polar coordinates is used to suggest that these maxima go to zero as r→0.
Define the function fx,y
Context Panel: Assign Function
fx,y=x⁢sin⁡x⁢y2−cos⁡x−cos⁡y→assign as functionf
Change to polar coordinates
F≔fr cosθ,r sinθ = r⁢cos⁡θ⁢sin⁡r2⁢sin⁡θ⁢cos⁡θ2−cos⁡r⁢cos⁡θ−cos⁡r⁢sin⁡θ
Find the extrema of F on circles of fixed radius r
The equation dFdθ=0 is complicated enough that solutions for θ^r cannot be found explicitly. Hence, Fr,θ^r is not available.
Instead, consider the animation in Figure 3.2.26(a) where the animation slider controls the value of r in a graph of Fr,t. The animation suggests that as r→0, the local extrema in the graph of Fr,t also approach zero.
Alternatively, approximate the numerator and denominator with Taylor polynomials, and compute the bivariate limit of the resulting rational function.
Figure 3.2.26(a) Animation in r for Fr,θ
Since sinu≐u−u3/3!, and cosu≐1−u2/2!+u4/4!, the approximating rational function is
for which the bivariate limit at the origin is
limit−13⁢x2⁢y⁢x2⁢y2−6x2+y2,x=0,y=0 = 0
Since the bivariate limit of f at the origin is 0, the required extension is
Indeed, limitfx,y,x=0,y=0 = 0.
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