Q. 29 - 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| < $\epsilon$.

$\underset{x\to 3}{\lim }(x+5)=8$

Step-by-Step Solution

Answer

$\delta =\epsilon$

Step1. Given information.

We have been given a limit statement as $\underset{x\to 3}{\lim }(x+5)=8$.

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

Step 2. Use algebra

From the given limit statement, we can identify

$$\begin{gathered} f\left(x\right)=x+5 \\ c=3 \\ L=8 \end{gathered}$$

$\mathrm{For} \epsilon >0$

$$\begin{gathered} \left|\right(x+5)-8|<\epsilon \\ |x+5-8|<\epsilon \\ |x-3|<\epsilon \end{gathered}$$

$\mathrm{For} 0<|x-c|<\delta , \mathrm{we} \mathrm{get} |x-3|<\epsilon$

Therefore, $\delta =\epsilon$