Tricky multivariable limit

Question

Calculate the limit without using L'Hopital's rule: $\lim_{(x,y) \rightarrow (0,0)} \sqrt{(x^2+y^2)} ~\log|y|$

Answer 1

The limit does not exist.
I show two paths that give different limits:

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Path 1. Consider $(x(t), y(t)) = (0, t)$ for $0 < t < 1$.
The function in consideration simplifies to $|t|\ln t$ and it is standard to see that this limit is $0$. (Can be done without L'H.)

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Path 2. Consider $(x(t), y(t)) = \left(\dfrac{1}{\ln t}, t\right)$ for $0 < t < 0.5$.

The function in consideration simplifies to $$-\sqrt{1 + t^2(\ln t)^2}.$$ The limit of this is $-1 \neq 0$.