<s>[INST] How to solve this limit 2 variables $\lim_{(x,y) \to (4,1)} \frac{y \sqrt x - 2y - \sqrt x + 2}{4 - x + x \sqrt y - 4 \sqrt y}$ Please anybody can help me solve this?
$$\lim \frac{y \sqrt x - 2y - \sqrt x + 2}{4 - x + x \sqrt y - 4 \sqrt y}$$
with $(x,y) \to (4,1)$
Thank you!
 [/INST] $$\frac{y\sqrt x-2y-\sqrt x+2}{4-x+x\sqrt y-4 \sqrt y}=\frac{\sqrt x (y-1)-2(y-1)}{\sqrt y(x-4)-(x-4)}=\frac{(y-1)(\sqrt x-2)}{(x-4)(\sqrt y-1)}=\frac{\sqrt y+1}{\sqrt x+2}$$
You can show that the function $f(x,y)=\frac{\sqrt y+1}{\sqrt x+2}$ is continuous at the point $(x,y)=(4,1)$ :
$$|f(x,y)-f(4,1)|=|\frac{\sqrt y+1}{\sqrt x+2}-\frac{1}{2}|=|\frac{2\sqrt y+2-\sqrt x-2}{2(\sqrt x+2)}|\leq \frac{2|\sqrt y-1|+|2-\sqrt x|}{2(\sqrt x+2)}<\epsilon$$ for $\|(x,y)-(4,1)\|_{l^2}$ small enough.
So the limit really exists and it is $f(4,1)=\frac{1}{2}$
</s>