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Proof of the limit f(x) = 1/x is 1/3 near 3

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Homework Statement


f(x) = 1/x
Proof that the limit of f(x) = 1/3 near 3
the answer provided by the text is: (1/x - 1/3) < e by requiring abs(x-3) < min(6e, 1)
I can see that the text has the right process and therefore it reaches the right answer, but I have done it in a different process which I believe is also correct but to a different answer. Can you see where have I made the mistake in my process?

Homework Equations





The Attempt at a Solution


This is my process:
Note:
(1/x - 1/3) = (3-x)/3x

Take
abs(x-3) < 1
=> 2 < x < 4
=> 6 < 3x < 12
=> 1/6 > 1/3x > 1/12
=> abs(1/3x) > 1/12 since 3x > 0

abs( (3-x) / (3x) ) < e
=> abs( (3-x) ( 1/3x) ) < e

Therefore, (1/12)(abs(3-x)) = (1/12)(abs(x-3)) < abs(3-x)abs(1/3x) < e
shows that abs(x-3)/12 < e
or abs(x-3) < e/ 12

therefore, my final answer becomes
(1/x - 1/3) < e by requiring abs(x-3) < min(e/12, 1)

Homework Statement





Homework Equations





The Attempt at a Solution

 

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