Talk:Calculus/Formal Definition of the Limit

I note what seems to be an error on the page. The author uses as an example in the beginning f(x)=x2 and then states that for this function the delta value can be obtained by taking the square root of a desired epsilon.

By my understanding of the author's definition of the limit this is not correct. To illustrate:

I am given an epsilon of .002 - this means that the delta values that I plug in will be allowed to vary the result f(x) by plus or minus .002. Given the example, attaining the limit of f(x) @ 2, this means to me that the delta I use would have to produce a maximum positive value for f(x) of 4.002. Working backwards, this would require an x value of 2.0005 meaning that the positive delta would be .0005 but taking the square root of the original epsilon I receive .045 - these two values do not coincide.

To verify, if I accept the square root of epsilon as producing my delta then f(2.045)=4.18203 - far outside the variance of the original epsilon.

I haven't been able to determine what the proper formula would be to derive the proper delta from a given epsilon in this case, but I thought it necessary to point this out.

--Lxman (talk) 15:51, 24 July 2010 (UTC)


 * You're right. Using $$\epsilon=0.002$$, the rule $$\delta=\sqrt\epsilon$$ would imply that choosing $$\delta=0.044$$, which is slightly less than $$\sqrt\epsilon$$, should satisfy the definition of the limit.  That is, for every $$\epsilon>0$$, if $$0<|x-2|<0.044$$, then $$0<|x^2-4|<0.002$$.  Let $$x=2.043$$.  Then $$0<|2.043-2|=0.043<0.044$$, but $$0<|2.043^2-4|=0.173849\nless0.002$$.


 * I derived an appropriate delta function as example 5 in Calculus/Choosing delta, and I updated the text of Calculus/Formal Definition of the Limit. Thanks for pointing it out!  --Greenbreen (discuss • contribs) 16:27, 23 March 2011 (UTC)

Subjective views
I removed the phrase "some of the most brilliant mathematicians" from the first paragraph because this is subject to opinion. There is no brilliant-ometer. Wikipedia should try to be factual, not biased. 98.194.121.63 (discuss) 20:11, 9 July 2012 (UTC)