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Problem 84

Proof
(a) Given that

$$\lim _{x \rightarrow 0}(3 x+1)(3 x-1) x^{2}+0.01=0.01$$ prove that there exists an open interval $(a, b)$ containing 0 such that $(3 x+1)(3 x-1) x^{2}+0.01>0$ for all $x \neq 0$ in $(a, b) .$

(b) Given that $$\lim _{x \rightarrow c} g(x)=L,$$ where $L>0,$ prove that there
exists an open interval $(a, b)$ containing $c$ such that
$g(x)>0$ for all $x \neq c$ in $(a, b) .$

$a )(-\delta, \delta)$
$b )(c-\delta, c+\delta)$

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