Problem

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

02:53

Problem 55 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.
$$
f(x, y)=\tan ^{-1}\left(\frac{|x|+|y|}{x^{2}+y^{2}}\right)
$$

Answer

$\frac{\pi}{2}$

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Calculus 3

Calculus 1 / AB

Thomas Calculus

Chapter 14

Partial Derivatives

Section 2

Limits and Continuity in Higher Dimensions

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Video Transcript

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'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

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