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Limit of a Function of two variables

Limit of a two variable Function|IIT JAM 2016 |Problem 6


Find the value of \lim\limits_{(x,y) \to (2,-2)} \frac{\sqrt{x-y}-2}{x-y-4} is

  • 0
  • \frac14
  • \frac13
  • \frac12

Key Concepts


Function of Several Variables

Limit of a function of several variables

Check the Answer


Answer: \textbf{(B)} \frac14

IIT JAM 2016 (Problem 6)

Function of several variables: Fleming, Wendell H

Try with Hints


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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Limit of a two variable Function|IIT JAM 2016 |Problem 6


Find the value of \lim\limits_{(x,y) \to (2,-2)} \frac{\sqrt{x-y}-2}{x-y-4} is

  • 0
  • \frac14
  • \frac13
  • \frac12

Key Concepts


Function of Several Variables

Limit of a function of several variables

Check the Answer


Answer: \textbf{(B)} \frac14

IIT JAM 2016 (Problem 6)

Function of several variables: Fleming, Wendell H

Try with Hints


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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