Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.'

share|cite|improve this question
here is a link to the notes:… page 2 Example 17.1.2. – student101 Sep 27 '12 at 8:24
up vote 2 down vote accepted

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

share|cite|improve this answer
yes i agree with what you wrote; but how do i know abs(y-2) <1 – student101 Sep 27 '12 at 8:30
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.