$\lim_{(x,y)\to(0,0)}\frac{xy}{\sqrt x+\sqrt y}$ considering domain $\{(x,y) \in \mathbb{R}^2 : x,y \ge 0, (x,y) \ne 0\}$
I tried using polar coordinates, but the theta function is unbounded. I also tried using the sandwich theorem, but could not find appropriate bounds. How do I approach this question
How do you find the multivariable limit $\lim_{(x,y)\to(0,0)}\frac{xy}{\sqrt x +\sqrt y }$
limitsmultivariable-calculus
Best Answer
As "geetha290krm" said: For $x>0,y>0$, you can rewrite as below $$\frac{xy}{\sqrt x +\sqrt y } =\frac 12 \frac{2xy}{\sqrt x +\sqrt y }\\0 \le\frac 12(\frac{xy}{\sqrt x +\sqrt y } +\frac{xy}{\sqrt x +\sqrt y } )=\frac 12(\frac{y\sqrt{x^2}}{\sqrt x +\sqrt y } +\frac{x\sqrt{y^2}}{\sqrt x +\sqrt y })\le \\\frac 12(\frac{y\sqrt{x^2}}{\sqrt x } +\frac{x\sqrt{y^2}}{\sqrt y })=\frac 12 (y\sqrt x+x \sqrt y)\\=\frac 12 \sqrt {xy}(\sqrt y+\sqrt x) $$ and it works for every path near $(0,0)$.