<s>[INST] Functions and inequalities I have no idea about this question. Please give a hand whoever can. I am a beginner.
Let $f(x)= x^2 - 10$.
Given $\epsilon > 0$, find the value "$a$"  so that $$|x -3 |< \min\{2,a\} \implies |f(x)-10|<\epsilon.$$
 [/INST] Assume that $x \neq 3$ to begin with:
$|f(x) - 10| = |x^2 - 10 - 10| = |x^2 - 20| < \epsilon \iff 20 - \epsilon < x^2 < 20 + \epsilon$. 
Now let $t = x - 3$, then $x = t + 3$. So:
$20 - \epsilon < (t+3)^2 < 20 + \epsilon$. Thus:
$\sqrt{20 - \epsilon} < |t+3| < \sqrt{20 + \epsilon}$. Thus:
$-\sqrt{20 + \epsilon} < t + 3 < \sqrt{20 + \epsilon}$, and:
$t + 3 < -\sqrt{20 - \epsilon}$ or $t + 3 > \sqrt{20 - \epsilon}$. So:
$-3 - \sqrt{20 + \epsilon} < t < -3 + \sqrt{20 + \epsilon}$, and:
$t < -3 - \sqrt{20 - \epsilon}$ or $t > -3 + \sqrt{20 - \epsilon}$. Thus:
$-3 - \sqrt{20 + \epsilon} < t < -3 - \sqrt{20 - \epsilon}$ or $-3 + \sqrt{20 - \epsilon} < t < -3 + \sqrt{20 + \epsilon}$. So to give one such "positive" $a$ ( $a > 0$ ), we can take: 
$a = \text{min} \{-3 + \sqrt{20 + \epsilon},3 - \sqrt{20 - \epsilon}\}$.
Note that it is possible for $a$ to be positive if you have a big enough $\epsilon$. 
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