Question

please reply me answer for this question​

Answer 1

SOLUTION

GIVEN

$\displaystyle f(x, y) = \frac{xy}{ {x}^{2} + {y}^{2} }$

TO DETERMINE

$\displaystyle \lim_{x \to 0} \: \lim_{y \to 0} \: f(x, y)$

CONCEPT TO BE IMPLEMENTED

If a function f(x, y) in some neighborhood of (a, b) then the limit

$\displaystyle \: \lim_{y \to b} \: f(x, y)$

if exists , is a function of x, say g(x). If the limit

$\displaystyle \lim_{x \to a} \: g(x)$

exists and equal to L then we write

$\displaystyle \lim_{x \to a} \: \lim_{y \to b} \: f(x, y) = L$

and say L is the repeated limit of f as

y → b, x → a

EVALUATION

Here it is given that

$\displaystyle f(x, y) = \frac{xy}{ {x}^{2} + {y}^{2} }$

Now

$\displaystyle \lim_{x \to 0} \: \lim_{y \to 0} \: f(x, y)$

$= \displaystyle \lim_{x \to 0} \: \lim_{y \to 0} \: \: \: \frac{xy}{ {x}^{2} + {y}^{2} }$

$= \displaystyle \lim_{x \to 0} \: 0$

$= 0$

$\therefore \: \: \displaystyle \lim_{x \to 0} \: \lim_{y \to 0} \: f(x, y) = 0$

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