Prove that $\lim_{x\to 2}x^2=4$ using $\epsilon-\delta$ definition.

By the mean of $\epsilon-\delta$ definition, $|x-2|\le \delta,|x+2|\le \delta+4$

then $|x-2||x+2|\le \delta(\delta+4),|x^2-4|\le \delta^2+4\delta$.

Assign $\epsilon=\delta^2+4\delta,|x^2-4|\le\epsilon$.


Is this method correct? If yes, why I always see people do this by putting $\delta = \min(1,\epsilon/5)$...?


  • 1
    $\begingroup$ You must fix $\epsilon$ at the begining, that's why you see much convolution to choose a $\delta$ that will do the job. Remember, that looks like $\forall \epsilon > 0, \exists \delta > 0, \ ...$ $\endgroup$ – Stop hurting Monica Mar 14 '13 at 13:52
  • $\begingroup$ Here is a related problem. $\endgroup$ – Mhenni Benghorbal Mar 14 '13 at 14:02
  • $\begingroup$ Then can I find the $\delta$ by solving $\delta^2+4\delta-\epsilon=0$? $\endgroup$ – A. Chu Mar 14 '13 at 14:24
  • 1
    $\begingroup$ @jasoncube: Follow the technique in the problem I referred you to it and you do not have to do all these calculations. $\endgroup$ – Mhenni Benghorbal Mar 14 '13 at 20:04

This is pretty good, but this goes the other way around.

You fix $\epsilon>0$ first.

Then you it suffices to find $\delta>0$ such that $\delta^2+4\delta\leq\epsilon$. You need to make this explicit. So indeed, lots of people would do it like this. If you take a priori $\delta\leq 1$, then $$ \delta^2+4\delta\leq \delta+4\delta=5\delta. $$ So it suffices to have $\delta \leq\epsilon/5$. If $\epsilon/5\leq 1$, the choice $\delta =\epsilon/5$ works by the above estimate. If not, the choice $\delta=1$ works, still by the previous estimate.

That's why most people would take $$ \delta:=\min\left(1,\frac{\epsilon}{5}\right) $$ which works either way.


No that one won't work, as you say $\epsilon=\dots$

The definition says that for every $\epsilon>0$ there is a delta, so you stell need to proof, that with your setting of $\epsilon$ you still can reach every value in $(0,\infty)$.

If you set $\delta $ to something you don't get those problems.


As pointed out by others, you have to follow the structure of the $\epsilon$-$\delta$ criterion in your proof. Try to fit your proof into the following frame:

Let $\epsilon > 0$.

We set $\delta = \ldots$ [Find some suitable $\delta > 0$ depending on $\epsilon$.]

Let $x\in\mathbb{R}$ with $\left|x - 2\right| < \delta$.

Then ...

So $\left|x^2 - 4\right| < \epsilon$.


We have a fixed $\varepsilon \gt 0$ and we want to set $\delta \gt 0$ so that for every $\delta_0 \in (-\delta,+\delta)\;$ (i.e. $|\delta_0| \lt \delta$),

$\tag 1 4- \varepsilon \lt 4 + 4 \delta_0 + {\delta_0}^2 \lt 4 + \varepsilon$

Now if $\delta = min(1, \frac{\varepsilon}{5})$,

$\quad 4 \delta_0 + {\delta_0}^2 \le |4 \delta_0| + |\delta_0| = 5 \, | \delta_0| \lt \varepsilon $

so the inequality on the rhs of $\text{(1)}$ will be true for this setting of $\delta$.

Fortunately, this setting also works for the lhs of $\text{(1)}$:

We have

$\quad -4 \delta_0 \le 4 |\delta_0 | \lt 4 \delta \le \frac{4 \varepsilon} {5} \lt \varepsilon$

But then $-\varepsilon \lt 4 \delta_0 \lt 4 \delta_0 + {\delta_0}^2$.

Note: For the lhs we only need to set $\delta = \frac{\varepsilon}{5}$. I guess we were lucky that we started with the rhs!


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