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The Sandwich Theorem for functions of two variables states that if $g(x, y) \leq f(x, y) \leq h(x, y)$ for all $(x, y) \neq\left(x_{0}, y_{0}\right)$ in a disk centered at $\left(x_{0}, y_{0}\right)$ and if $g$ and $h$ have the same finite limit $L$ as $(x, y) \rightarrow\left(x_{0}, y_{0}\right),$ then$$\lim _{(x, y) \rightarrow\left(x_{0}, y_{0}\right)} f(x, y)=L$$Use this result to support your answers to the questions in Exercises $45-48 .$Does knowing that$$2|x y|-\frac{x^{2} y^{2}}{6}<4-4 \cos \sqrt{|x y|}<2|x y|$$tell you anything about$$\lim _{(x, y) \rightarrow(0,0)} \frac{4-4 \cos \sqrt{|x y|}}{|x y|} ?$$Give reasons for your answer.

2

Calculus 3

Calculus 1 / AB

Chapter 14

Partial Derivatives

Section 2

Limits and Continuity in Higher Dimensions

Applications of the Derivative

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