Find the value of $\lim\limits_{(x,y) \to (2,-2)} \frac{\sqrt{x-y}-2}{x-y-4}$ is
Function of Several Variables
Limit of a function of several variables
Answer: $\textbf{(B)} \frac14$
IIT JAM 2016 (Problem 6)
Function of several variables: Fleming, Wendell H
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]
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