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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.$$f(x, y)=\frac{2 x}{x^{2}+x+y^{2}}$$

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

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, 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's the solution.

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