Limits/continuity - finding largest delta

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SUMMARY

The discussion centers on finding the largest delta (δ) such that |x - 5| < δ implies |1/x - 1/5| < 1/100. The initial attempt incorrectly derived a lower bound for 1/x, leading to δ = 5/19. Upon reevaluation, the correct approach identifies an upper bound for 1/x, resulting in δ = 5/21, which satisfies the condition. This highlights the importance of correctly interpreting inequalities in limit and continuity problems.

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roman93
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Hey guys, I found this question and it started bugging me:

find the _largest_ δ such that |x - 5| < δ => |1/x - 1/5| < 1/100.

This is what I did to try solve the question:

From |1/x - 1/5| < 1/100 : I got 1/x > 19/100 and so I wanted x < 100/19
plugging that back into the first inequality it gave δ = 5/19

But I don't think what I did is right because when I was trying values for x which satisfied this it gave answers which were over 1/100.

So my question is how would I solve this type question.
Any help is very much appreciated :)

Thank you in advance!
 
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I think I know where I went wrong, what I should be doing is finding an upper bound for 1/x rather than a lower bound. The answer I now have is 5/21 and this seems to work.

If I have gone wrong somewhere, please do point it out =]
 

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