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How do we approach limitis involving absolute values in both numerator and denominator with multiple variables? For instance, if we have something like: $$ \lim_{(x,y)\to(0,0)} \frac{|x|}{|x|+|y|} $$

My guess is to use absolute value inequality, so we will have three cases when $$ x=0, x<0, x>0$$ and then check the limit when $$y=0, x=0, x=y$$ to see whether the limit exists or not. Is that correct?

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Guide:

Try to consider the limit along different path.

For example $y=x$, $y=2x$.

Remark about your strategy for general question:

There are path where $x$ and $y$ changes sign.

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