Q. 41 - Calculus

Question

For each limit statement, use algebra to find δ or N in terms of $\epsilon$ or M, according to the appropriate formal limit definition. 

 $\underset{x\to -2^{+}}{lim} (1+\sqrt{x+2})=1$ find δ in terms of $\epsilon$.

Step-by-Step Solution

Answer

$\delta ={\epsilon }^2$

Step1. Given information.

We have been given a limit statement as $\underset{x\to -2^{+}}{lim} (1+\sqrt{x+2})=1$.

We have to find δ in terms of $\epsilon$. 

Step 2. Use algebra.

From the given limit statement, we can identify 

$$\begin{gathered} f\left(x\right)=1+\sqrt{x+2} \\ c=-2 \\ L=1 \\ \mathrm{For} \epsilon >0 \\ \left|\right(1+\sqrt{x+2})-1|<\epsilon \\ \left|\sqrt{x+2}\right|<\epsilon \\ |x+2|<{\epsilon }^2 \\ \mathrm{For} 0<|x-c|<\delta , \mathrm{we} \mathrm{get} |x+2|<{\epsilon }^2 \\ \mathrm{Therefore}, \delta ={\epsilon }^2 \end{gathered}$$