Chapter 1.2, Problem 84E

Proof

(a) Given that

$\underset{x\to 0}{\lim }(3x+1)(3x-1)x^2+0.01=0.01$

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

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