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 1. Unfortunately, showing 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 on circles of radius are computed, and then if it can be shown that these maxima tend to as , it will have been established that the bivariate limit of at the origin is . To this end, express in polar coordinates and find the extrema for fixed as a function of .
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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Simplify the derivative .
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Solve the equation .
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Simplify the solutions subject to the restriction that .
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Evaluate at each solution of , thus obtaining the extreme values of on the circle of radius
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Obtain, as , the limiting value of the common extreme value of on the circle of radius
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Context Panel: Limit operator
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Expression palette: Evaluation template
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Since the bivariate limit of at the origin is 1, the required extension is
Indeed, = .