Q. 30 - Calculus

Question

For each limit statement , use algebra to find δ > 0 in terms of $\epsilon$ > 0 so that if 0 < |x − c| < δ, then | f(x) − L| < $\epsilon$.

$\underset{x\to -2}{\lim }(4-2x)=8$

Step-by-Step Solution

Answer

$\delta =\frac{\epsilon }{2}$

Step1. Given information.

We have been given a limit statement as $\underset{x\to -2}{\lim }(4-2x)=8$.

We have to find $\delta \mathrm{in} \mathrm{terms} \mathrm{of} \epsilon$.

Step 2. Use algebra

From the given limit statement, we can identify 

$$\begin{gathered} f\left(x\right)=4-2x \\ c=-2 \\ L=8 \end{gathered}$$

$$\begin{gathered} \mathrm{For} \epsilon >0 \\ \left|\right(4-2x)-8|<\epsilon \\ |4-2x-8|<\epsilon \\ |-2x-4|<\epsilon \\ 2|x+2|<\epsilon \\ |x+2|<\frac{\epsilon }{2} \\ \mathrm{For} 0<|x-c|<\delta , \mathrm{we} \mathrm{get} |x+2|<\frac{\epsilon }{2} \\ \mathrm{Therefore}, \delta =\frac{\epsilon }{2} \end{gathered}$$