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Concept explainers
a.
To find: that x < 1 or x = 1.
a.
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Answer to Problem 35E
xmust be equal to one.
Explanation of Solution
Given information: Let x =0.99999.
Calculation:
Find,is x less than or equal to 1.
x =0.99999.
x appears to be less than one however it also appears to be infinitely close to 1.
The difference between 1 and x can only be determined by the value and place of the last decimal. Since the decimal repeats continuously there is no way to discern a difference between 1 and x . Because it is continuous x must be equal to 1.
x appears to be less than 1 however it also appears to be infinitely close to 1.
The difference between x and 1 can only be determined by the value and place of its last decimal, however the decimal is continuously repeating. This suggests that there can be no discernable difference between 1 and x.
Therefore x must be equal to one.
b.
To find: the value of x , sum a geometric series.
b.
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Answer to Problem 35E
The sum of the geometric series is 1 and that is value of x.
Explanation of Solution
Given information: Let x =0.99999.
Calculation:
x =0.99999.
So, the sum of the geometric series is 1 and that is value of x.
c.
To find: how much decimal representation does the number 1 have.
c.
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Answer to Problem 35E
Two.
Explanation of Solution
Given information: Let x =0.99999.
Calculation:
Since 0.9999=1. So 0.9999… and 1.000 are the two representations of 1.
d.
To find: which numbers have more than one decimal representation.
d.
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Answer to Problem 35E
All the numbers with infinitely many 9’s at the tail.
Explanation of Solution
Given information: Let x =0.99999.
Calculation:
Some examples:
0.1234599999…=0.12346.
0.0000000120001=0.000000012000099999…
So, all the numbers with infinitely many 9’s at the tail.
Chapter 8 Solutions
Single Variable Calculus: Concepts and Contexts, Enhanced Edition