Chapter 1.4, Problem 30ES

(a) Show that $\left|\frac{28}{3x+1}-4\right|=\left|\frac{12}{3x+1}\right|\cdot \left|x-2\right|$

(b) Is $\left|\frac{12}{\left(3x+1\right)}\right|$ bounded if $\left|x-2\right|<4$ ? If not, explain; if so, give a bound.

(c) Is $\left|\frac{12}{\left(3x+1\right)}\right|$ bounded if $\left|x-2\right|<1$ ? If not, explain; if so, give a bound.

(d) Fill in the blanks to complete a proof that $\underset{x\to 2}{\lim }\left[\frac{28}{3x+1}\right]=4$

Suppose that $\epsilon >0$ . Set $\delta =\min \left(1,\text{\_\_\_\_\_\_}\right)$ and assume that $0<\left|x-2\right|<\delta$ . Then $\begin{array}{c}\left|\frac{28}{3x+1}-4\right|=\left|\frac{12}{3x+1}\right|\cdot \left|x-2\right|\\ <\text{\_\_\_\_\_\_}\cdot \left|x-2\right|\\ <\text{\_\_\_\_\_\_}\cdot \text{\_\_\_\_\_\_}\\ \text{=}\in \end{array}$