I have to calculate (if it exists) the following limit: $$\lim_{(x,y)\rightarrow(0,0)}\frac{\ln(1+|xy|)}{x^{2}+y^{2}}.$$ What I did is simply to consider that $\ln(1+\theta)\sim\theta$ as $\theta\rightarrow 0$. So I have the limit: $$\lim_{(x,y)\rightarrow(0,0)}\frac{|xy|}{x^{2}+y^{2}}.$$ I know the inequality: $$|x|+|y|\ge \sqrt{x^{2}+y^{2}}$$ which I used in the previous exercises, but here there is a product of absolute values and not a sum, so I think it is not useful. How should I proceed?
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2You might use $2xy\le x^2+y^2$? This would give you an upper bound of $\frac 12$. – DominikS Nov 03 '23 at 11:12
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When your limit is a quotient of polynomials, you should look at the degrees. Here both numerator and denominator have the same degree $2$, which for a several-variable limit suggests it doesn't exist.
Now you can try $x=0$, where the limit is $0$, and $x=y$, where the limit is $1/2$.
Martin Argerami
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