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Chapter 1: Q. 22 (page 107)

Suppose fx=mx+bis a linear function with m鈮�0, and let c be any real member.

Part (a): Show that for all 鈭�>0, if 0<x-c<鈭�m, then fx-fc<鈭�.

Part (b): What does the implication in part (a) have to do with limits?

Part (c): Illustrate the implication in part (a) with a labeled graph. Explain in terms of slopes why it makes sense that given 鈭�>0, the corresponding未>0is未=鈭�m.

Short Answer

Expert verified

Part (a): To prove fx-fc<鈭�, put x=cand multiply mon every side and solve the inequation.

Part (b): Using limits, we get limx鈫�cx=c.

Part (c): The required graph is given below,

Step by step solution

01

Part (a) Step 1. Given information.

Consider the given question,

fx=mx+b鈭�>0

If 0<x-c<鈭�mthenfx-fc<鈭�.

02

Part (a) Step 2. Substitute c in x in the given function.

Substitute x=cin the given function,

fc=mc+c

For all 鈭�>0,

0<x-c<鈭�m

Multiply mon every side,

role="math" localid="1648275234533" 0路m<x-c路m<鈭�m路m0<mx-mc<鈭�0<mx+c-mc+c<鈭�0<fx-fc<鈭�

Therefore, fx-fc<鈭�.

Hence, proved.

03

Part (b) Step 1. To defined the given implication with limits.

From the definition of limit,

If 0<x-c<鈭�m,

Then, it can be written,

limx鈫�cx=c

04

Part (c) Step 1. Sketch the graph.

On sketching the graph, we get,

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Most popular questions from this chapter

For each limit statement, use algebra to find 未 or N in terms of 蔚or M, according to the appropriate formal limit definition.

limx鈫�鈭�2+鈥�(1+x+2)=1 find 未 in terms of 蔚.

Use what you know about one-sided limits to prove that a function f is continuous at a point x=cif and only if it is both left and right continuous at x=c.

Use algebra to find the largest possible value of 未 or smallest possible value of N that makes each implication true. Then verify and support your answers with labeled graphs.

Ifx>N, then1x2鈭�0<0.001

Each function in Exercises 9鈥�12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f.

limx鈫�2-f(x)=-鈭�,limx鈫�2+f(x)=鈭�,f(2)=3.

If fis a continuous function, what can you say aboutlimx鈫�1f(x)?
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