Q. 66 - 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.
$\underset{x\to -\infty }{\lim }(3x-5)=-\infty$

Step-by-Step Solution

Answer

The proof of given limit $\underset{x\to -\infty }{\lim }(3x-5)=-\infty$ is,  

$N=\frac{M+5}{3}$

Step 1. Given information

We are given, 

$\underset{x\to -\infty }{\lim }(3x-5)=-\infty$

Step 2. Writing the delta-epsilon proofs

The strategy is to use algebra to find N in terms of M,

From the given limit, we get

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

Glven $M<0$, choose $N=\frac{M+5}{3}$.

If $x<N$, we have,

$$\begin{gathered} 3x-5>3N-5 \\ =M \end{gathered}$$

So,

$$\begin{gathered} 3 N-5=M \\ 3 N=M+5 \\ N=\frac{M+5}{3} \end{gathered}$$