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

Thus, the limit does not exist because $\cos 2 \theta$ lies between $-1$ and $+1,$ depending on thevalue of $\theta .$

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 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's the solution.

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