Q. 12 - Calculus

Question

Sketch a labeled graph that illustrates what is going on in the proof of Theorem 1.6 in the reading. Your graph should include two different $ϵ$-bars and a graphical reason that they cannot overlap.

Step-by-Step Solution

Answer

Step 1. Given information.

We have to sketch a labeled graph that illustrates what is going on in the proof of Theorem 1.6 in the reading.

Step 2. Sketch the graph.

Theorem 1.6 says that for a limit to exist, if $\underset{x\to c}{lim} f\left(x\right)=L\text{ and }\underset{x\to c}{lim} f\left(x\right)=M\text{, then }L=M\text{. }$

For the first limit expression $\underset{x\to c}{lim} f\left(x\right)=L$, the graph is

For the second limit expression the graph is :