Epislon Delta Proof

[pessimist92]

When proving that limit (x->5) 1/x = 0.2, find the largestδ > 0 that works for ε = 0.0025 = 1/400

My attempt: | 1/x - 1/5 | = | (5-x)/5x | = | (x-5) /5x | < ε. | (x-5) /5x | < 1/400 whenever 0 < | x-5| < δ. So for some real number M, if 1/|5x| < M, then | (x-5) /5x | < M |x-5|. Hence | x-5| <ε/M = δ.

Now |x-5| < 1 implies 4<x<6 which further implies 1/30 < 1/|5x| < 1/20, thus M= 1/20 is suitable. So we can take δ = min{1,20*ε}. That is the largest δ is 20/400 = 0.05.

Is this argument correct?

 

[Crimson Sunbird]

The answer 0.05 is not correct – try \(\displaystyle x=5.06\): \(\displaystyle \left|\frac15-\frac1{5.06}\right|<0.0025\) whereas \(\displaystyle |5.06-5|>0.05\). So a greater value than 0.05 for \(\displaystyle \delta\) is possible.

I would go about the problem this way:

\(\displaystyle \displaystyle-\frac1{400}<\frac1x-\frac15<\frac1{400}\)​

\(\displaystyle \displaystyle\Leftrightarrow\ \frac{79}{400}<\frac1x<\frac{81}{400}\)

\(\displaystyle \displaystyle\Leftrightarrow\ \frac{400}{81}<x<\frac{400}{79}\)

\(\displaystyle \displaystyle\Leftrightarrow\ -\frac5{81}<x-5<\frac5{79}\)

Hence \(\displaystyle \displaystyle\delta_{\max}=\min\left\{\left|-\frac5{81}\right|,\left|\frac5{79}\right|\right\}=\frac5{81}\).

 

[pessimist92]

The answer 0.05 is not correct – try \(\displaystyle x=5.06\): \(\displaystyle \left|\frac15-\frac1{5.06}\right|<0.0025\) whereas \(\displaystyle |5.06-5|>0.05\). So a greater value than 0.05 for \(\displaystyle \delta\) is possible.

I would go about the problem this way:

\(\displaystyle \displaystyle-\frac1{400}<\frac1x-\frac15<\frac1{400}\)​

\(\displaystyle \displaystyle\Leftrightarrow\ \frac{79}{400}<\frac1x<\frac{81}{400}\)

\(\displaystyle \displaystyle\Leftrightarrow\ \frac{400}{81}<x<\frac{400}{79}\)

\(\displaystyle \displaystyle\Leftrightarrow\ -\frac5{81}<x-5<\frac5{79}\)

Hence \(\displaystyle \displaystyle\delta_{\max}=\min\left\{\left|-\frac5{81}\right|,\left|\frac5{79}\right|\right\}=\frac5{81}\).

Oh I get it. Now I understand how to do computational epsilon-delta problems. Thanks for your help!