Problem

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

02:58

Problem 50 Easy Difficulty

Continuous extension Define $f(0,0)$ in a way that extends
$$
f(x, y)=x y \frac{x^{2}-y^{2}}{x^{2}+y^{2}}
$$
to be continuous at the origin.

Answer

$f(0,0)=0$

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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

So for this question asked us to define F 00 in a way that extends on this value of the function to be continuous at the origin. So look at this. In order for a function at three continuous at the origin, the function value at the origin must equals a limit of the function as X Y approaches origin. So that means basically, the F 00 is going to equal to the limit. X y goes to 00 of my function of X y. So therefore, we have to define f 00 b this limit, like I just wrote. And so what we do next is to find this limit. However, I can't just easily find us alone right now, So I kind of want to use the same serum that we just learned, and some the previous problems, something someone needs some kind of absolute about hears. I'm going to go ahead and find the limit. Sex. Why goes to 00 was absolute value of my function, because then I could get some kind of value here, preferably a less than Abel to a great ankle, to using some tricks. And then I can use sandwich there to determine what this value will be. So I think that's a value that means I'm gonna have a value of X Y. Times expert minus y squared over x squared plus y squared the limit, unable to limit as X y goes to 00 of when I go ahead and just separate this little bit someone of X y over x squared plus y squared, which I dropped the absolute value Serino inside here is positive since we have two squares added together. Time's up, sir. Value of X cred minus y squared. Something that we have seen before is that I have some Valium extents. Why is less than equal to 1/2 of X squared plus y squared? So using that, that means this limits going less than or equal to limit of disfunction someplace X. Why was this 1/2 X squared plus y squared? Just let me listen. Angle to limit X Y ghosts here. Zero These two things cancel. So I'm just left with this 1/2 next mice y squared, which is equal to zero. So we see that some not all up that the limit says X y goes to 00 of the absolute value of F of X Y is less than or equal to zero. And so, since his limit is the absolute value, though, I also know that the limit as X Y goes to 00 of my function has been greater than equal zero since I'm taking a limit of the absolute value of this. And so then, bye sandwich there. Um Lim as X y ghost 00 after that's why is it zero? And this further implies that the limit his ex wife goes to 00 ofjust, the function is equal to zero, so we're going to find that's 00 t equals zero.

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