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Chapter 1: Q. 19 (page 98)

Write each limit in Exercises 19鈥�42 as a formal statement involving 未,蔚,N,and/orM,and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

limx鈫�-3x+7=2

Short Answer

Expert verified

For all epsilon positive, there exists a delta positive such that if,

x鈭�(-3-未,-3)鈭�(-3,-3+未)Thenx+7鈭�(2-蔚,2+蔚).

The graph is

Step by step solution

01

Step 1. Given information.

The given expression islimx鈫�-3x+7=2.

02

Step 2. Explanation.

From the given expression, we have, c=-3,L=2.

The limit expression can be written as a formal statement as below,

For all epsilon positive, there exists a delta positive such that if,

localid="1648209826708" x鈭�(-3-未,-3)鈭�(-3,-3+未)Thenx+7鈭�(2-蔚,2+蔚).

03

Step 3. Graph the expression. 

The graph illustrating the roles of the constants is shown as below:

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Most popular questions from this chapter

For each limit statement , use algebra to find 未 > 0 in terms of 蔚> 0 so that if 0 < |x 鈭� c| < 未, then | f(x) 鈭� L| < 蔚.

limx鈫�0(x3+1)=1

For each limit statement , use algebra to find 未 > 0 in terms of 蔚> 0 so that if 0 < |x 鈭� c| < 未, then | f(x) 鈭� L| < 蔚.

limx鈫�31x=13;youmayassume未鈮�1

For each limit limx鈫�cf(x)=Lin Exercises 43鈥�54, use graphs and algebra to approximate the largest value of 未such that ifx鈭�(c-未,c)鈭�(c,c+未)thenf(x)鈭�(L-蔚,L+蔚)

limx鈫�-1x2-2x-3x+1=-4,蔚=0.1

Write each of the inequalities in interval notation:

fx-L<鈭�

For each limit statement in Exercises 41-44, use algebra to find 未or Nin terms of 蔚or M, according to the appropriate formal limit definition.

limx鈫�1-11-x=鈭�, find未in terms ofM.
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