in-exercises-51-56-find-the-limit-of-f-as-x-y-rightarrow00-or-show-that-the-limit-does-not-exist-f-4

Question

In Exercises $51-56,$ find the limit of $f$ as $(x, y) \rightarrow(0,0)$ or show that the limit does not exist.
$$
f(x, y)=\frac{2 x}{x^{2}+x+y^{2}}
$$

Uma Kumari · Accepted Answer

In this problem, you have to find the limit of the given function as X ray approaches 00 or to show that the limit does not exist on the given function is f x y Z equals toe to X over X squared plus X plus y square. Now to change to polar coordinates, X is equals toe are assigned pita. Why is equals tow R sine theta and our approaches zero that gives us Lim X Y approaches 00 two x over X squared plus X plus y square Now substituting the values of X and Y you could get limit our approaches zero toe are assigned Feta Ola, our assigned Rita whole square plus our assigned pita plus our scientific data whole square. Now simplifying this we will get limit Our approach is zero to our assigned data or what are square co sign square pita plus sine square pita plus are consigned pita. Now we know that sign Square feet are plus co sign Square theater is equals to one Now, By using this, we can write limit our approaches. Zero who are chosen Tita over r squared, plus our cozy in pita simplifying this we will get limit out of approaches. Zero to cosign pita over our plus for science Tita. Now applying the limit, we will get to costs scientific data over zero plus cause scientific data is equals toe to toe scientific data over cause I in Tita. Therefore the limit does not exist for chosen Tita is equals zero. So that&#x27;s the solution.

Uma Kumari · Answer

in this problem, you have to find the limit off the Cuban function as exploit approaches 00 or show that limit does not exist. Now the given function is and X y is equals toe white square over X square plus y square. Now to change to polar coordinates. X is it was toe are assigned. Tita, Why is equals who are signed, Tita and our approaches zero We will get limit X y approaches 00 by square, over X square, plus by square Now substituting in values off X and y we get limit. Our approach is zero r sine theta full square. Hola. Our assigned data Whole square plus r sine theta all square. By simplifying this, we will get limit our coaches zero our square sine squared pita all I are Square Science Square Peter Plus who? Zion Square Theater. Now we know that Science Square Theater plus co Science Square Thika is equals to one. Therefore, by using this, we will get limit. Our approach is zero sign square Trita. Now we get financial air signs square Tita. Therefore the limit does not exist because sign square theater is between zero and one depending on Tita. So there&#x27;s the solution

Uma Kumari · Answer

in this problem, we have to find a limit off as X y approaches 00 or to show that the limit does not exist. Now we are given f o X y z equals toe X square minus y squared over X squared plus y square. Now we change to polar coordinates. X is equals toe are for science, Tita and bicycles to our science, Tita and our approaches Zero now, from this right limit, X y approaches 00 X square minus y squared or x squared plus y square, Substituting the values off X and Y we get limit our approach to zero Our assigned Rita Whole square minus our scientific. A whole square or our core scientists who square plus our scientific, the whole square. Now simplifying this limit. Our approach is zero our square for science square. Trita miners are square science. Where Rita Oh, are our square for Science square, Tita. Plus our square science. Where Peeta. Now for the simplifying you get limit. Our approach is zero are square CropScience square, Peter minus sign square. Peter, Over our square for Science Square. Peter plus Science Square. Peter, we know that Science Square theta plus Co. Zion Square Theater is equals toe one. Therefore, from using this where I limit our approach is zero ho science squared theta minus science Where Tita further we get limit. Our approach is zero who assigned to theta and by applying limit bigger assigned to theta. Therefore, the limit does not exist because yeah, assigned to Tita Nice between minus one and last one, depending on value All Tita So that&#x27;s the solution.

Uma Kumari · Answer

it is given that a four picks comma by is equals to excuse minus X into y squared, divided by X squared, plus by square, they will use polar coordinates to solve it. So it&#x27;s as equals toe are in Tokyo. Scientific data. Why is equals to our into science? Data on our tends to zero, so we have lim X comma via tends to zero comma. Zero X cubed minus X into y squared, divided by X squared, plus five square substitute in the value eggs and why we get limit. Our tends to zero are in because I in Tita whole Q minus are in Tokyo. Santita into are into sine sine theta whole square, divided by are into cozy in data. Whole square plus are into $70 where it would be equals toe limit. Our tends to zero Cuban to cosign cubed feta minus. I&#x27;ll cube in to cosign pita into science square pita divided by our square in the coastline square theta plus our square into Science Square. Tita, which would be equals to limit cartons to zero our Cuban to co sign Cube data minus goes I in data in two sine squared data divided by it. Our square in to co sign square pita plus signs. Well, Peter, it would be equals to limit our tends to zero are in Do co sign cuchi Tita minus co Scientific data into signs square pita. Since Science Square three topless school Science Square three ties equals to one. Upon applying the limits, we get zero does. The required solution is zero. This is done so

Uma Kumari · Answer

in this problem, we have to find a limit off F as X y approaches 00 or show that the limit does not exist. Now the given function is f ex lie is equals toe engine in birth, more of eggs, plus mortal. Why over X square plus by square. Now change to polar coordinates. X is equals to our scientific data. And why is it was tow our scientific data and our approaches. Zero, we&#x27;ll get limit. X y approaches 00 ancient in MERS model X plus more of I. Oh, uh, X squared plus y square now substituting values off X and y then our is greater than zero. We get limit. Our approach is zero positive. The engine in bars, more off our assigned Tita plus model are signed. Dita Oh, uh, our assigned pita. Whole square, plus our science. Tita. Whole square. Now we know that model are assigned. Tita is equals toe under rule. Our square science were pita, which is our go Zion pita and model. Our scientific tha is equals toe under route. Our square science square pita is equals. Tow our science, Peter. Therefore, by using this, they can write limit. Our approach is zero positive. 10 gent in verse are chosen. Pita plus our sign, people over our square was nine square teeter plus our square sign Square theater? No. In the next step limit. Our approach is zero positive enjoyment in hers are Cosin pita plus sign, Peter, or what are square co Science Square? Dita less science. Where? Dita. Now we know that sign. Square Peter Plus or Science Square. Peter, this was +21 Now, from this we can write Limit our approaches. Zero positive engine tin birds cause I in pita plus sign pita over. All right. Therefore, by further simplifying and applying the limit value gate danger in verse. Coarsen Tita, the sine theta over zero, which gives us engine in verse in for night. And we know that injured universe infinite is by over two. Now, when our is less than zero, we get limit. Our approach is you negative engine in birth. More off our cousin Tita, plus more off our scientific data. Or what? Our design, Tita. Whole square plus are signed. Pita, whole square. Now you know that model are assign Tita is equals toe a new router. Our square assigned square pita, which is it was toe are assigned more off our science. Pita is equals toe. Another route are square Science square pita which is equals. Tow our science data. Now. By using this, we will right. Limit. Our approach is zero negative and joint Inverse are assigned Pita plus our sign Peter, over our square co sign square Tita, plus our square sine squared pita. Now for the simplifying it limit our approaches, you know, Negative engine in verse. Our assigned data plus sign Peter over our square co sign square pita plus sine squared pita. Now we know that sign. Square teeter less. Oh, science, where Peter is equals to one by using this will simplify limit Our approach is zero negative and gent in verse cause I in Tita less science Dita over are now applying limit. We will get 10 geant in birth cause I in Tita plus sign pita over zero, which is equals toe engine it in verse. Infinite. On Ben Genting Waas infinite is by over two. Therefore the required solution is by over two

Uma Kumari · Answer

in this problem, you have to find the limit of the given function as X y approaches 00 or show that the limit does not exist. Now we&#x27;re given that f x y is equals toe or sign X cubed. Minus right. You over X squared plus y squared. Now change to Paul. The coordinates will take. X is equals toe are assigned. Tita, why is equals? Tow our science, Tita and our approaches. Zero. This becomes limit X y approaches 00 or sign. X cube minus y cube over X squared plus y square. Now substituting in values off X and y, we will get limit. Our approach is zero Hussein, our assigned. Tita. All to minus our sign. Pita, How will you? Over our science, Tita. Whole square. Plus our scientific data. Whole square. We will get limit. Our approach is zero co sign. Are you for signed? You Tita, Minus argue. Sign you, Tita, Over our squares. Cosign square, Tita. Plus our square science square, Tita, now taking are as common limit art for zero co sign. Are you consigned You, Peter? Minus sign you pita over our square cause I in squared theta minus science. Where data. Now we know that sine squared Peter plus cosigned square theater is equals to one. Therefore, by using this will gay limit our approach is zero or sign are cause signed. You, Tita Minus sign you Tita. Now applying the limit we get Cosin Tita is equals to one. Therefore the required solution is born Mhm.

Problem

In Exercises $51-56,$ find the limit of $f$ as $(…

Problem 54 Hard Difficulty

In Exercises $51-56,$ find the limit of $f$ as $(x, y) \rightarrow(0,0)$ or show that the limit does not exist.
$$
f(x, y)=\frac{2 x}{x^{2}+x+y^{2}}
$$

Answer

the limit does not exist for $\cos \theta=0$

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the limit does not exist for $\cos \theta=0$

Thomas Calculus

Lily A.

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Vectors Intro

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

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Heather Z.

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

Thomas Calculus