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Suppose I have $\epsilon > 0$ and $\delta$ is min of $\left\{1,\frac{\epsilon}{10}\right\}$

If $0< |x-1| < \delta$, is it then true that $|3x-3|< \epsilon$? I am saying yes and here is why.

I can deduce that that $0<|3x-3|<3\delta$ , and since $\delta$ is min of $\left\{1,\frac{\epsilon}{10}\right\}$, then $3\delta$ is min of $\left\{3,\frac{3\epsilon}{10}\right\}$

Now $0<|x-1|< 1$ which leads to $0<|3x-3|< 3$,

or $0< |x-1|<\frac{\epsilon}{10}$, which leads to $0< |3x-3|<\frac{3\epsilon}{10}$

now I conclude that $|3x-3|<\epsilon$, because obviously $\frac{3\epsilon}{10}<\epsilon$

and I realize I have a problem because 3 does not have to be less than $\epsilon$. Suppose $\epsilon$ was 0.1.

So this means my original statement is false.

Michael Hardy
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stackdsewew
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  • Are you trying to use the fundamental definition of the limit of a function? – Mhenni Benghorbal Apr 12 '16 at 04:44
  • This isn't pertaining to any particular function, bur rather asking if it is always true. I have edited it because a portion of the question was not appearing on the screen. – stackdsewew Apr 12 '16 at 04:47
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    You are correct that $3$ does not have to be less than $\epsilon$, however, $|3x-3|$ is still less than $\epsilon$ by the argument you gave (the reason this works is that $|x-1|$ may be less than $1$ and indeed can be arbitrarily close to $0$. – Justin Benfield Apr 12 '16 at 05:12
  • you have simply $3 \delta=$ $3 \min (1,\epsilon /10);\leq$ $ 3 (\epsilon /10)<\epsilon.$ – DanielWainfleet Apr 12 '16 at 06:14

1 Answers1

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If $\epsilon =0.1$, then $\delta =\min \{ 1,\frac{0.1}{10}\} =\frac{0.1}{10}=\frac{1}{100}$.

Now we suppose that $0<|x-1|<\frac{1}{100}$ and ask:

Is $|3x-3|<0.1$?

The answer is yes because:

$|x-1|<\frac{1}{100}\Rightarrow 3|x-1|<\frac{3}{100}\Rightarrow |3x-3|<\frac{3}{100}$

and $\frac{3}{100}<0.1$, thus $|3x-3|<\epsilon$ as required.

Justin Benfield
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  • Thanks. I'm also going to post another similar question, sorry to beat a dead horse. I just want to really understand this thing. – stackdsewew Apr 12 '16 at 05:22
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    That's actually the right thing to do with anything that isn't making complete and total sense in mathematics. Keep investigating it until it does make sense. – Justin Benfield Apr 12 '16 at 05:24
  • I have posted it. http://math.stackexchange.com/questions/1738709/epsilon-delta-verification – stackdsewew Apr 12 '16 at 05:34