Limit of a Function of two variables

So, this problem has double limit. But we will not consider this first i.e., first try to solve the above limit without seeing $(x,y) \to (2,-2)$ , just like we used to do in our school days!!!

Here also we cannot put $x,y$ directly, as it will become undefined.

Now,

$\lim\limits_{(x,y) \to (2,-2)} \frac{(x-y)-4}{(x-y-4)(\sqrt{x-y}+2)}$ [Multiplied by $\sqrt{x-y}+2$ on numerator and denominator ]

Now again, try to simplify the above limit !!!

After simplifying,

We have,

$\lim\limits_{(x,y) \to (2,-2)} \frac{1}{\sqrt{x-y}+2}$ [ Since $(x,y) \to (2,-2) \Rightarrow x-y-4 \ne 0$]

Now in this part we will use the limits, i.e., $(x,y) \to (2,-2)$

$\Rightarrow \frac{1}{\sqrt{2+2}+2}=\frac14$[ANS]

So, this problem has double limit. But we will not consider this first i.e., first try to solve the above limit without seeing $(x,y) \to (2,-2)$ , just like we used to do in our school days!!!

Here also we cannot put $x,y$ directly, as it will become undefined.

Now,

$\lim\limits_{(x,y) \to (2,-2)} \frac{(x-y)-4}{(x-y-4)(\sqrt{x-y}+2)}$ [Multiplied by $\sqrt{x-y}+2$ on numerator and denominator ]

Now again, try to simplify the above limit !!!

After simplifying,

We have,

$\lim\limits_{(x,y) \to (2,-2)} \frac{1}{\sqrt{x-y}+2}$ [ Since $(x,y) \to (2,-2) \Rightarrow x-y-4 \ne 0$]

Now in this part we will use the limits, i.e., $(x,y) \to (2,-2)$

$\Rightarrow \frac{1}{\sqrt{2+2}+2}=\frac14$[ANS]