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Find the limit of $f$ as $(x, y) \rightarrow(0,0)$ or show that the limit does not exist.$$f(x, y)=\frac{x^{3}-x y^{2}}{x^{2}+y^{2}}$$

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Calculus 3

Chapter 14

Partial Derivatives

Section 2

Limits and Continuity in Higher Dimensions

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12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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02:34

Find the limit of $f$ as $…

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All right so for this problem we can first consider the absolute value of the function now using the triangle inequality. We can say that this is less than equal to the tale to this part being pulled up from x cubed. So we have the absolute lit of x that we have x, squared over x, squared plus y square, which is of course less than equal to 1. Posity square is greater than equal to 0.2. Can capture that i just had any 1 there, and then we also have again plus the absolute value of x, this time coming from the minus x factor with y squared and second at end of f and again y squared is equal to y squared plus x. Squared so y squared over y square plus x squared is thus equal to 1. So, of course, this is twice absolute value: xativa 10 to 0. This value, of course, also tends to 0 since x 0 replies absolute x, 0 implies twice to 0 as well. So we conclude that in the limit, the absolute value of f 0- that is at the limit of the limit we desire is 0 again- is x, y 10 to 0 p.

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