The requisite extension assigns to the origin the value of the bivariate limit of at the origin. Hence, what is required is to show that this limit is 0. Unfortunately, showing 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 on circles of radius are computed, and then shown to tend to as . At best, a graph of in polar coordinates is used to suggest that these maxima go to zero as .
Define the function
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Context Panel: Assign Function
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Change to polar coordinates
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Find the extrema of on circles of fixed radius
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The equation is complicated enough that solutions for cannot be found explicitly. Hence, is not available.
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Instead, consider the animation in Figure 3.2.26(a) where the animation slider controls the value of in a graph of . The animation suggests that as , the local extrema in the graph of also approach zero.
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Alternatively, approximate the numerator and denominator with Taylor polynomials, and compute the bivariate limit of the resulting rational function.
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module()
local q,p;
q:=r*cos(theta)*sin(r^2*cos(theta)*sin(theta))/(2-cos(r*cos(theta))-cos(r*sin(theta)));
p:=plots:-animate(plot,[q,theta=0..2*Pi],r=0..1,frames=11,digits=2);
print(p);
end module:
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Figure 3.2.26(a) Animation in for
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Since , and , the approximating rational function is
for which the bivariate limit at the origin is
=
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Since the bivariate limit of at the origin is 0, the required extension is
Indeed, = .