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

[hotsmart123]

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)

 

[mighty2000]

Your process is also correct, but the issue is in the last step where you take the minimum of e/12 and 1. This is not necessary and can lead to a slightly different answer. Instead, you should take the minimum of e and 12, which will give the same result as the text's answer. This is because the absolute value of x-3 can never be less than 1, so taking the minimum with e/12 is unnecessary.

Therefore, the correct final answer would be (1/x - 1/3) < e by requiring abs(x-3) < min(e, 12).

In general, when taking the minimum in a limit problem, it is important to consider the bounds of the variable and make sure that the value you are taking the minimum with is actually achievable. In this case, abs(x-3) can never be less than 1, so taking the minimum with e/12 is not necessary.