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Question Number 196399 by tri26112004 last updated on 24/Aug/23

$$a/\underset{(x,y)\to (0,2)}{\lim }{(1+xy)}^{\frac{2}{x^2+xy}}$$$$b/\underset{(x,y)\to (0,0)}{\lim }(x^2+y^2)sin\left(\frac{1}{xy}\right)$$$$c/\underset{(x,y)\to (\mathrm{\infty },\mathrm{\infty })}{\lim }(x^2+y^2)e^{-(x+y)}$$

Answered by MM42 last updated on 24/Aug/23

$$a){lim}_{(x,y)\to (0,2)}{\left[{(1+xy)}^{\frac{1}{xy}}\right]}^{\frac{2xy}{x^2+xy}}$$$${lim}_{(x,y)\to (0,2)}{\left[{(1+xy)}^{\frac{1}{xy}}\right]}^{\frac{2y}{x+y}}=e^2$$$$$$$$b)-(x^2+y^2)⩽sin\left(\frac{1}{xy}\right)⩽(x^2+y^2)$$$$\Rightarrow {lim}_{(x,y)\to (0,0)}-(x^2+y^2)⩽{lim}_{(x,u)\to (0,0)}sin\left(\frac{1}{xy}\right)⩽{lim}_{(x,y)\to (0,0)}(x^2+y^2)$$$$\Rightarrow Ans=0$$$$c)ifx=y>0\Rightarrow {lim}_{x\to +\mathrm{\infty }}\frac{2x^2}{e^{2x}}=0$$$$ifx=y<0\Rightarrow {lim}_{x\to +\mathrm{\infty }}2x^2{e}^{2x}=\mathrm{\infty }$$$$$$$$$$

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