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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),$$$\sqrt{x^{2}+y^{2}}<\delta \Rightarrow|f(x, y)-f(0,0)|<\epsilon$$$$f(x, y)=(x+y) /\left(x^{2}+1\right), \quad \epsilon=0.01$$

$\epsilon=0.01$

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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consider that the function a prolific school move I is equals to explosive by upon X squared plus one and a positive riel number absolutely is equals to 0.1 objective is to prove that their existence their town it is greater than zero such that for all X come alive, then under root X Square plus by square. It's smaller than their town, which gives us a four x coma by minus F of zero comma. Zero. Hold more small than Delta, then find a number. Delta is greater than zero for which under root X square plus by square A small than they come. We get I folks. X goma y my news fo zero comma zero full moon It's smaller than absolute 0.1 has considered a four x Goma y minus fo zero comma zero it's equals two explosive by upon X squared plus one minus zero Plan zero. Suppose you're a place fan whole mood which is equals toe explosive by upon X squared plus one my new zero whole mode. But simplifying it we get explicit by upon X squared plus one which is smaller than equals to do into and rude X squared, plus by square scenes. Express by Morris. Smaller than equals toe to in 200 x squared plus y squared they are four f o x Goma y minus fo zero comma. Zero hold more a smaller than equals toe do into under rude X squared, plus by square. So choose Delta is equal to absolutely upon to which is equals to 0.5 on four under Route X Squared plus y square. It's smaller than Delta Be gig F o X common by my news f of zero comma zero full mode is smaller than equals toe to into under Group X Squared plus y squared, which is smaller than toe into delta. It is equal to two into absolute. Divided by two things, Delta is equals to absolute by two hands, there is data which is equal to 0.5 which is created in zero, such that for all take school. Moviein means that under root X Square plus by square, it's smaller than 0.5 That gives us f o X coma by minus fo zero comma zero whole mood Hey smaller than 0.1

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