<s>[INST] Find $\lim\limits_{(x,y)\to(1,-1)}{\frac{e^{x-y}\tan{(x+y)}}{x^2-y^2}}$ I need to find $\lim\limits_{(x,y)\to(1,-1)}{\frac{e^{x-y}\tan{(x+y)}}{x^2-y^2}}$. I came up with this:
Since $x^2-y^2$ changes its sign, I can'y apply $\tan{(x+y)}\sim(x+y)$ just yet. So,$\lim\limits_{(x,y)\to(1,-1)}{|\frac{e^{x-y}\tan{(x+y)}}{x^2-y^2}|}=\lim\limits_{(x,y)\to(1,-1)}{\frac{e^{x-y}|\tan{(x+y)}|}{|x^2-y^2|}}=\lim\limits_{(x,y)\to(1,-1)}{\frac{e^{x-y}|x+y|}{|x-y|.|x+y|}}=\lim\limits_{(x,y)\to(1,-1)}{\frac{e^{x-y}}{|x-y|}}=\frac{e^2}{2}$ 
Is this correct and is there a better way to approach the problem?
 [/INST] We have $$\frac{e^{x-y}\tan(x+y)}{x^2-y^2}=\frac{e^{x-y}}{x-y}\cdot\frac{\sin(x+y)}{x+y}\cdot\frac{1}{\cos(x+y)}$$
Break this into three parts using composition of limits.
Part 1:
Set $z(x,y)=x-y$ so $z(x,y)\to 2$ as $(x,y)\to(1,-1)$. Then $$\lim\limits_{(x,y)\to(1,-1)}\frac{e^{x-y}}{x-y}=\lim\limits_{z\to 2}\frac{e^{z}}{z}=\frac{e^2}{2}$$
Part 2:
Set $z(x,y)=x+y$ so that $z(x,y)\to 0$ as $(x,y)\to(1,-1)$. Then
$$\lim_{(x,y)\to(1,-1)}\frac{\sin(x+y)}{(x+y)}=\lim\limits_{z\to 0}\frac{\sin(z)}{z}=1$$
Part 3:
Set $z(x,y)=x+y$ again so that $z(x,y)\to 0$ as $(x,y)\to(1,-1)$. Then
$$\lim\limits_{(x,y)\to(1,-1)}\frac{1}{\cos(x+y)}=\lim\limits_{z\to 0}\frac{1}{\cos(z)}=\frac{1}{\cos(0)}=1$$
Conclusion:
We have $$\lim\limits_{(x,y)\to(1,-1)}\frac{e^{x-y}\tan(x+y)}{x^2-y^2}=\frac{e^2}{2}\cdot 1\cdot 1=\frac{e^2}{2}$$
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