<s>[INST] Verify the Limit: $\lim_{x\rightarrow 9} (\sqrt{x} +9) = 12$ I am using the definition of the limit of a function. So far I have: 
$$|f(x)-L|<\epsilon \iff |(\sqrt{x}+9)-12|<\epsilon \iff |\sqrt{x}-3|<\epsilon.$$ 
This is where I get stuck. Any help is appreciated. 
 [/INST] Keep in mind that in deriving these proofs we are always working backwards.  We start with $\epsilon$ and try to find $\delta$ that will make it true.
Let $\epsilon > 1$.  Since $\epsilon $ may be as small as we like we may as well assume $1 > \epsilon > 0$.  I'm saying this now, because later we will have a reason for this.
So we want to find a $\delta$ (expressed in terms of $\epsilon$) so that whenever $|x - 9|< \delta$ we will have to have $|\sqrt{x} - 3|< \epsilon$.
We can find this $\delta$ one of two ways:
Forward:
$|x - 9|<\delta \implies$
$-\delta < x - 9 < \delta \implies $
$9 -\delta < x < 9+\delta \implies $
$\sqrt{9-\delta} < x  <\sqrt{9+\delta}\implies$
Now if we can somehow find some way that $3 - \epsilon \le \sqrt{9-\delta}$ and then $\sqrt{9+\delta} \le 3+ \epsilon$ we'll be on our way.
So we want $3-\epsilon \le \sqrt{9 -\delta}$ so $9 - 6\epsilon + \epsilon^2 \le 9- \delta$.  So we need $\delta \le 6\epsilon - \epsilon^2$.
And we want $\sqrt {9-\delta} \le 3 + \epsilon$ so $9-\delta \le 9 + 6\epsilon +\epsilon^2$.  So we need $-6\epsilon -\epsilon^2 \le \delta$. Well, that's true for any positive $\delta$.
How can we find a $\delta$ where both will be true.  Well, if we assume $0 < \epsilon < 1$, then $\epsilon^2 < \epsilon$ so if we can define $\delta$ so that $\delta \le 5\epsilon < 6\epsilon - \epsilon^2$ we'd be done.
So define $\delta = 5\epsilon$.  If $|x - 9| < \delta$ then $-5\epsilon < x-9 < 5\epsilon$ and $9-6\epsilon + \epsilon^2 < 9-5\epsilon < x < 9+5\delta < 9 + 6\delta + \delta^2$.  So $ 3-\epsilon < \sqrt {9-5\epsilon} < \sqrt x< \sqrt{9+5\delta} < 3 + \epsilon$. So $-\epsilon < \sqrt x < \epsilon$ and $|\sqrt x - 3| < \epsilon$.
The proof is done.
Or backwards:
$|\sqrt{x} - 3| <\epsilon $ is implied by
$-\epsilon < \sqrt{x} - 3 <\epsilon \Leftarrow$
$3-\epsilon < \sqrt{x} < 3 + \epsilon\Leftarrow$
$9-6\epsilon +\epsilon^2 < x < 9 + 6\epsilon + \epsilon^2 \Leftarrow$
$-6\epsilon +\epsilon^2 < x -9 < 6\epsilon + \epsilon^2 \Leftarrow$
$-5\epsilon < -6\epsilon +\epsilon^2  < x-9 < 5\epsilon < 6\epsilon + \epsilon^2$.
Note: $\Leftarrow$ means is implied by.  We are going in the opposite direction then we are used to.
Normally we say $x -9 < 5\epsilon \implies x-9 < 5\epsilon < 6\epsilon + \epsilon^2\implies x< 6\epsilon + \epsilon^2$ because we are saying: if A then B.  
But $\Leftarrow$ means:  A can come about from B. Or: if B then A.  $x-9 < 5\epsilon \Leftarrow x-9 < 6\epsilon + \epsilon^2$.
$\Leftarrow |x-9|< 5\epsilon$.
S if we let $\delta = 5\epsilon$ we are done.
So define $\delta = 5\epsilon$.  If $|x - 9| < \delta$ then $-5\epsilon < x-9 < 5\epsilon$ and $9-6\epsilon + \epsilon^2 < 9-5\epsilon < x < 9+5\delta < 9 + 6\delta + \delta^2$.  So $ 3-\epsilon < \sqrt {9-5\epsilon} < \sqrt x< \sqrt{9+5\delta} < 3 + \epsilon$. So $-\epsilon < \sqrt x < \epsilon$ and $|\sqrt x - 3| < \epsilon$.
The proof is done.
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