each-of-exercises-59-62-gives-a-function-fx-y-and-a-positive-number-epsilon-in-each-exercise-show--2

Question

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 $(x, y),$
$$
\sqrt{x^{2}+y^{2}}<\delta \Rightarrow|f(x, y)-f(0,0)|<\epsilon
$$
$$
f(x, y)=y /\left(x^{2}+1\right), \quad \epsilon=0.05
$$

Uma Kumari · Accepted Answer

in this problem, we are given that f o x y is equals toe by our X squared plus one and a positive real number at Salon is it was toe 0.5 and we have to show that Delta is greater than zero for all such that X Goma y and Underwood X squared plus y square is less than data and more off fo x y minus. F 00 is less than Absalon. Now Let their basic was 20.5 Then more off X is less than data and more of why is less than data We get model fo x y minus fo 00 is equals toe model bio our X squared less one minus zero. We get more off by or X square plus one, please less than equal do more off. Why less than 0.0 is equals toe epsilon. Therefore, the solution is epsilon is equals tau 0.5

Uma Kumari · Answer

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&#x27;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&#x27;s smaller than absolute 0.1 has considered a four x Goma y minus fo zero comma zero it&#x27;s equals two explosive by upon X squared plus one minus zero Plan zero. Suppose you&#x27;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&#x27;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&#x27;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

Uma Kumari · Answer

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&#x27;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

Uma Kumari · Answer

it is given that a forfeit Skomal is equals. Toe explains why divided by two plus co sign it let mhm their ties equals to 0.1 then goes in. It is greater than equals two minus one and smaller than equals to one, which gives us two plus co sign. Eggs is greater than equals to one and smaller than equals 23 we get one upon three is smaller than equals to one upon to place cause Hynix Elevator smaller than equals toe one. It gives us explosive by more. Divided by three, they&#x27;re smaller than equals toe explosive. I divided by two place for cynics. Whole mode, which is smaller than equals toe explosive by mood, which is also smaller than equals traits. More plus fine work, then make smaller, smaller than Delta on by Mode is also smaller than that and gives us a full fix coma by minus fo zero comma zero whole more is equals. Toe explains why divided by bless goes Heinicke minus zero hole more, which is equals to express by divided by two plus co cynics. Whole morgue well, which is smaller than equals toe, takes more plus five more. It would also be smaller than 0.1 plus 0.1 which is equals True zero going 02 which is equals to excellent. There&#x27;s the solution is absolutely is equals. Toe 0.2

Uma Kumari · Answer

it has given that a four x com Abika mercy is equals. Toe X squared plus y squared, plus the square leg Delta is equals toe under 0.0 and five. So under rude eggs where plus y squared plus C square the smaller than data it gives us F o x com Abiko Matzzie minus fo zero comma zero comma zero whole mode is equals toe X square, plus by square, plus the square. My new zero hold more it would be equals two x squared plus y squared plus C square. Holy more We get on the road a square plus y squared, plus the square who&#x27;s square a smaller than under Route 0.0 Gunfight whole square. It would be equals to 0.15 which is equals to absolute does. The solution is absolute is equals to 0.15 This is the final answer

Uma Kumari · Answer

it is given that f o x com Abiko Matzzie, This equals two x into my embassy lead. Delta is equals to zero point toe. Then X motor, smaller than Delta by mode, is also smaller than data on Z mode is also smaller than data. We get a four x com Abiko matzzie minus a 40 comma. Zero comma zero whole mode is equal to a four x comma by commas e minus zero. Hold more. It would be equals toe extent to buy into the mode be good x mode and to bi mode and to see more, which is smaller than zero point to hold. But simplifying it we get 0.8 which is equals to absolutely does. The solution is absolutely as equals to 0.8 This is the final answer.

Problem

Each of Exercises $59-62$ gives a function $f(x, …

Problem 60 Medium Difficulty

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 $(x, y),$
$$
\sqrt{x^{2}+y^{2}}<\delta \Rightarrow|f(x, y)-f(0,0)|<\epsilon
$$
$$
f(x, y)=y /\left(x^{2}+1\right), \quad \epsilon=0.05
$$

Answer

$\epsilon=0.05$

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$\epsilon=0.05$

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George B. Thomas, Maurice D. Weir, Joel Hass, Frank R. GiordanoThomas Calculus

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George B. Thomas, Maurice D. Weir, Joel Hass, Frank R. Giordano

Thomas Calculus