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how to determine all points of a continuous function f x y sin x y x y limit tending to 1 if x y is different from 0 if x

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How to determine all points of a continuous function f (X,y)={sin(X+y)/X+y Limit tending to 1 If X+y is different from 0 If X+y =0

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Answer to a math question How to determine all points of a continuous function f (X,y)={sin(X+y)/X+y Limit tending to 1 If X+y is different from 0 If X+y =0

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f(x,y) = \begin{cases} \frac{\sin(x+y)}{x+y}, & \text{if } x + y \neq 0 \\\lim_{t \to 1} \frac{\sin t}{t}, & \text{if } x + y = 0 \end{cases}

1. For points where

x + y \neq 0

, the function is given by

f(x,y) = \frac{\sin(x+y)}{x+y}

.

2. For points where

x + y = 0

, we evaluate the limit:

\lim_{t \to 1} \frac{\sin t}{t} = \frac{\sin 1}{1}

.

Therefore, the continuous function f(x,y) is

f(x,y) = \begin{cases} \frac{\sin(x+y)}{x+y}, & \text{if } x + y \neq 0 \\\frac{\sin 1}{1}, & \text{if } x + y = 0 \end{cases}

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