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By considering different paths of approach, show that the functions in Exercises $35-42$ have no limit as $(x, y) \rightarrow(0,0) .$$$f(x, y)=\frac{x^{4}-y^{2}}{x^{4}+y^{2}}$$

Hence, for $k=0$ along the $x$ -axis $y=0$ and the limit is 1 .Thus, $f(x, y)$ have different limits for different values of $k$ .Hence, the limit of the function $f(x, y)$ does not exist.

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, consider the function F X y physicals, toe extra, about four minus y square or actually the power four plus twice where. How the objective is to show that the function has no limit by considering different parts of approach on the limit cannot be found by direct substitution, which gives the indeterminant form that will be 0/0. Now examine the limit off the function along the different parts. That is why is equals to K X square so that so take the limit as limit X Y approaches 00 alone. Why is equals toe K X Square, which gives us f x y Z equals toe limit. X y approaches 00 along. Why is it was to be expire X to the power four minus vice square over. It's about four. Bless my square now, putting the values or pie, we will get limit X approaches zero X to the power four minus K X square, Whole square over. Excited about four plus a X Square whole square now, But simply find this. We will get limit. X approaches zero X to the path forward one minus games. Where over extra about four bunless is where we will get one minus case where over one plus is where therefore the value off the limit depends on the value off constant K. Hence the value off limit is wearing with the value off. Hey, now for K is equals to one the value X fight approaches 00 along the parabola. Why is it was two x square and the limit is limit. X y approaches 00 along Why is equals two x square f x y is It was toe one minus one whole square over one plus one whole square because her key is it was 21 By solving this really get zero over to that is it was 20 Hence for K is equals to one that along the parabola why is equal to X square and the limit is zero. Now we'll ask poor he is equals to zero the value exp I approaches 00 along the x X is Why is it was 20 and the limit will be limit. X y approaches 00 along why is equals to zero f x y is equals toe one minor, zero whole square over one plus zero whole square. Because here K is equals to zero by solving the civilian one or one is equals to one hand, or K is equals to zero. Along the X axis is why is equals to zero and the limit is one. Therefore, the function F x y have different limits or different values. Uh, yeah, hands the limit off function. F X Y does not exist, so that's the solution.

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