Maple determines the bivariate limit at the origin to be 1.
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Context Panel: Assign Function
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Context Panel: Evaluate and Display Inline
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Context Panel: Limit (Bivariate)
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To see why the bivariate limit at the origin might have the value 1, divide the numerator and denominator of to put into the form
The rational functions in the numerator and denominator of this form of both tend to zero as . Indeed, Maple provides the following bivariate limits, obtained either through the Context Panel, or by application of the limit command with the appropriate syntax.
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Auxiliary Limit (1)
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Auxiliary Limit (2)
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Clearly, then, the bivariate limit of as will necessarily be 1, in which case the required extension is
Analytic justification for Auxiliary Limit (1) is based on the following estimate.
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Inequality 3, Table 3.2.1
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Inequalities 4, 6, and 7, Table 3.2.1
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Consequently, , and the limiting value of zero is established.
Analytic justification for Auxiliary Limit (2) is based on the following estimate.
Consequently, , and the limiting value of zero is established.
Indeed, = .