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Chapter 1: Q. 6 (page 97)

Find punctured intervals on which the function fx=1xlnx+2is defined, centered around

ax=0bx=-1cx=-1.5

Short Answer

Expert verified

Part (a). The punctured intervals is-0.5,1鈭�0,0.5.

Part (b). The punctured intervals is-1.5,-1鈭�-1,-0.5.

Part (c). The punctured intervals is-2,-1.5鈭�-1.5,-1.

Step by step solution

01

Step 1. Given information. 

The given function isfx=1xlnx+2.

02

Part (a) Step 1. Calculation.

For x=0

The punctured interval c-未,c鈭�c,c+未 is given by:
0-0.5,0鈭�0,0+0.5=-0.5,1鈭�0,0.5

03

Part (b) Step 1. Calculation.

For x=-1

The punctured interval c-未,c鈭�c,c+未is given by:

-1-0.5,-1鈭�-1,-1+0.5=-1.5,-1鈭�-1,-0.5

04

Part (c) Step 1. Calculation.

For x=-1.5

The punctured interval c-未,c鈭�c,c+未 is given by:
-1.5-0.5,-1.5鈭�-1.5,-1.5+0.5=-2,-1.5鈭�-1.5,-1

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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�12x4=2;youmayassume未鈮�1.

Write delta-epsilon proofs for each of the limit statements limx鈫�cfx=Lin Exercises 47-60.

limx鈫�23x2-12x+15=3.

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)=2,limx鈫�2+f(x)=1,f(2)=1.

Write a delta鈥揺psilon proof that proves that fis continuous on its domain. In each case, you will need to assume that 未 is less than or equal to 1.

f(x)=x3

Write delta-epsilon proofs for each of the limit statements limx鈫�cfx=Lin Exercises 47-60.

limx鈫�12x+4=6.
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