#How to calculate limit of this multi variables function:

13 messages · Page 1 of 1 (latest)

cursive void
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The function is in the image.
I've tried calculate lim (x, 0) -> (0, 0) , and lim (0, y) -> (0, 0), the result are 0.
But when I calculated lim (x, x) -> (0, 0) and lim (y, y) -> (0, 0), the result are both 1/2.
Which answer is right, and is there any explaination? Thanks!

image.png
white juncoBOT
cursive void
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<@&286206848099549185> please help, thanks!

woeful kiln
cursive void
woeful kiln
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Second, is the question to find the limit, or to check whether the limit exists?

cursive void
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Yeah, as I described in title, find the limit

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The answer is there are no limit icant

woeful kiln
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It is [the case that there is no limit]!

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I've tried calculate lim (x, 0) -> (0, 0) , and lim (0, y) -> (0, 0), the result are 0.
But when I calculated lim (x, x) -> (0, 0) and lim (y, y) -> (0, 0), the result are both 1/2.

#

That shows why for you, if your limit is
$$
\lim_{(x,y) \to (0,0)} \frac{xy}{x^{2} + y^{2}}
$$

lean mantleBOT
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chartbit

788085606483361802.png
woeful kiln
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You approach from different directions and get different values for the limit, so the limit can't exist!