Q. 62 - Calculus

Question

For each of the limit statements in Exercises 61-66, write a $\delta -M,N-ϵ$, or $N-M$ proof, according to the type of limit statement.
$\underset{x\to -2^{-}}{\lim }\frac{1}{x+2}=-\infty$

Step-by-Step Solution

Answer

Delta-epsilon proof is,  

$\delta =\frac{1}{M}$

Step 1. Given information

We are given, 

$\underset{x\to -2^{-}}{\lim }\frac{1}{x+2}=-\infty$

Step 2. Writing the delta-epsilon proofs

The strategy is to use algebra to find $\delta$ in terms of M,

From the given limit, we get

$$\begin{gathered} f\left(x\right)=\frac{1}{x+2} \\ c=-2 \\ L=-\infty \end{gathered}$$

Given $M<0$, choose $\delta =\frac{1}{M}$.

If $x\in (-2-\delta ,-2)$, we have $0<x+2<\delta$,

Now for $\delta >0$,

$$\begin{gathered} |x-(-2\left)\right|<\delta \\ |x+2|<\delta \\ \frac{1}{|x+2|}<\frac{1}{\delta }<M \end{gathered}$$

Hence, 

$\delta =\frac{1}{M}$