relation between repeated limit & double limit

[Sarita Math]

SORRY for one mistake.

If the double limit exist, it's not necessary that the repeated limits will also exist. But if they exist, they all are equal.

Also, if the repeated limit exists and are not equal or any one of them fails to exist, then the double limit does not exist.

f(x,y)=xsin(1/y)+ysin(1/x)

repeated limits does not exist, but the double limit exists.

So, Aman is right. I just check that right now.

Extremely sorry for the mistake.

[Naman Mitruka]

maam can u please show how to write the steps to solve this

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[Sarita Math]

Here,

Similarly, 

Hence the repeated limits exist and they are equal.

Again, since there are points arbitrarily near (0,0) at which f is equal to 0 and points arbitrarily near (0,0) at which f is equal to 1. Therefore, for any ,

|f(x,y)-f(0,0)=|f(x,y)| 

for all points in any neighborhood of (0,0).

Hence,

    does not exist.