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Chapter 1: Q. 10 (page 97)

Use interval notation to fill in the blanks that follow. Your

answers will involve未,蔚,N,and/orM.

Iflimx鈫�鈭�f(x)=-鈭�then for allrole="math" localid="1649070353117" M>0there is somerole="math" localid="1649070361631" N>0 such that ifx鈭�_____then鈭�_____.

Short Answer

Expert verified

If limx鈫�鈭�f(x)=-鈭�then for all M>0there is some N>0such that if x鈭�N,鈭�thenf(x)鈭�-鈭�,M.

Step by step solution

01

Step 1. Given information. 

The given function islimx鈫�鈭�f(x)=-鈭�.

02

Step 2. Explanation.

From the given expression, we have, c=鈭�,L=-鈭�.

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

For all M>0, there is someN>0 such that if,

x鈭�N,鈭�thenf(x)鈭�-鈭�,M.

03

Step 3. Statement.

Therefore, statement will be written as:

If limx鈫�鈭�f(x)=-鈭�then for allM>0, there is some N>0such that if,

x鈭�N,鈭�thenf(x)鈭�-鈭�,M.

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Most popular questions from this chapter

For each limit in Exercises 33鈥�38, either use continuity to calculate the limit or explain why Theorem 1.16 does not apply.

limx鈫�3x4.

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�12x4=2;youmayassume未鈮�1.

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For each limit in Exercises 33鈥�38, either use continuity to calculate the limit or explain why Theorem 1.16 does not apply.

limx鈫�-5(3x-2).

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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