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Each of Exercises $59-62$ gives a function $f(x, y)$ and a positive number $\epsilon .$ In each exercise, show that there exists a $\delta>0$ such that for all $(x, y),$$$\sqrt{x^{2}+y^{2}}<\delta \Rightarrow|f(x, y)-f(0,0)|<\epsilon$$$$f(x, y)=x^{2}+y^{2}, \quad \epsilon=0.01$$

$|f(x, y)-f(0,0)|<0.01=\epsilon$

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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in this problem, we are given fo X y is equals toe under rude X Square, plus Why square and a positive riel number Exile in is equals to 0.1 and we have to show that Delta is greater than zero for all such that X comma, vile and under rude X Square plus Vice Square is less than data and moored off fo x Y minus fo 00 is less than a silent now lead days. It was 20 point what we take under root X squared plus y square less than that now, which gives us under X squared plus y square less than 0.1. We can also write this as X squared plus y square less than 0.1 by taking square both sides. So we have a model F o X y, minus fo 00 more is equals toe model X squared. That's why square minus zero less than 0.1 which gives us mortal echoplex life minus fo 00 less than 0.0, while equally toe of silence. Therefore, the solution is model f of X y, minus fo 00 less than 0.1 is it was toe Obziler

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