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For the limit $$\lim_{x\to 5}\sqrt{x-1}=2$$ find a $\delta>0$ that works for $\epsilon=1$.

In another words, find a $\delta>0$ such that for all $x$, $$0<|x-5|<\delta \implies |\sqrt{x-1}-2|<1$$


Ok so here's what I did... $$|\sqrt{x-1}-2|<1$$ $$-1<\sqrt{x-1}-2<1$$ $$1<\sqrt{x-1}<3$$ $$1<x-1<9$$ $$2<x<10$$
Since I have to get $x$ in the simplified inequality above to change to $x-5$, I decided to subtract 5 on all sides like this: $$2-5<x-5<10-5$$ $$-3<x-5<5$$

But I don't know if that's right or not. Please help?

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  • $\begingroup$ Try it out! Plug in different values for your $\delta$ from the range you found. $\endgroup$
    – The Count
    Jun 27, 2016 at 19:48

2 Answers 2

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Assuming your algebra is correct (I didn't check it carefully), you have shown that $|\sqrt{x-1}-2|<1$ if and only if $-3<x-5<5$. Now you want to find $\delta>0$ so that if $|x-5|<\delta$ then $-3<x-5<5$. But $|x-5|<\delta$ is the same as $-\delta<x-5<\delta$. So what is a suitable value of $\delta$?

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  • $\begingroup$ Yes, that's right $\endgroup$ Jun 27, 2016 at 21:13
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You know that if $-3 < x-5 < 3 \implies -3 < x-5 < 5$. This means you can take $\delta = 3$.

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  • $\begingroup$ Yea the answer is 3, but how did you directly get -3 < x - 5 < 3 from -3 < x - 5 < 5? $\endgroup$ Jun 27, 2016 at 21:11
  • $\begingroup$ It comes from $-3$, then I think the other is $3$. $\endgroup$
    – DeepSea
    Jun 27, 2016 at 21:14
  • $\begingroup$ Can't we take $-5$ and $5$, instead of $-3$ & $3$? $\endgroup$ Jul 31, 2018 at 2:08
  • $\begingroup$ @MrReality No because we want a tight bound to imply a looser bound. i.e. $-3<x-5<3$ implies $-3<x-5<5$. But a loose bound doesn't imply a tight bound. i.e. $-5<x-5<5$ doesn't imply $-3<x-5<5$ (take $x=1$ for example). $\endgroup$
    – Jam
    Nov 16, 2019 at 17:42

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