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context: currently learning about 'definition of limits' as applied to functions of 2 variables.

$f(x,y)= xy $ consider if: $(x,y)-> (1,2)$

In finding the limit using 'definition of limits'; my notes then write the "statement": $|y|<3$ if $|y-2|<1$.

I agree that the above statement, by itself is correct, however how is it deduced? As in how did the finding of the limit lead to the inferences in the 'statement.'

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here is a link to the notes: rutherglen.science.mq.edu.au/~maths/Chen-notes/lnfycfolder/… page 2 Example 17.1.2. –  student101 Sep 27 '12 at 8:24

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up vote 2 down vote accepted

Without words: $$ |y-2|<1 \Rightarrow |y| \leq |y-2|+2 < 3. $$

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yes i agree with what you wrote; but how do i know abs(y-2) <1 –  student101 Sep 27 '12 at 8:30
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I guess it is a typical "condition without loss of generality". Since $y \to 2$, you can always consider those values of $y$ that live in some neighborhood of $2$. Here the neighborhood is the interval $(1,3)$. –  Siminore Sep 27 '12 at 9:40

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