This is from a calculus textbook (self-study), but my problem is not with the calculus material itself. I feel I'm missing something (obvious!) from algebra. In any case:(adsbygoogle = window.adsbygoogle || []).push({});

The problem statement

Find an appropriate number L (a limit), and a deleted neighborhoodNofa, such that a givene> 0,

L - e < f(x) < L + e when x is in N.

Exercise I can solve

I can solve most of the exercises, but I can't figure out some of them. Here's one I can solve:

1. f(x) = 5 - 3x, a = 2

The answer in the back of the book, which is my answer, is:

L = -1, 0 < | x - 2 | < e/3

My work is, L = 5 - 3*2 = 5 - 6 = -1

L - e < 5 - 3x < L + e

-1 - e < 5 - 3x < -1 + e

-6 - e < -3x < -6 + e

(-6 - e)/-3 > x > (-6 + e)/-3

2 + e/3 > x > 2 - e/3

e/3 > x - 2 > -e/3

0 < | x - 2 | < e/3 (to delete x=2)

Exercise I can't solve

(or at least can't get the answer in the back of the book):

3. f(x) = 4/x, a = 2

The answer in the back of the book is:

0 < | x - 2 | < 2e/(2 + e)

Here's how far I get:

L = 4/2 = 2

2 - e < 4/x < 2 + e

-e < 4/x - 2 < e

-e/4 < x - 1/2 < e/4

Now, here is where I'm missing something. I can either delete 1/2 (instead of 2),

0 < | x - 1/2 | < e/4

or, if I subtract 3/2 instead,

-e/4 - 3/2 < x - 2 < e/4 - 3/2

I can't unify the leftmost and rightmost terms because they don't have the same absolute value.

Trying really hard to get the book's answer, working from one of the above equations, I can multiply by 8/(2+e):

-e/4 < x - 1/2 < e/4

(8/(2+e)) * (-e/4) < (8/(2+e)) * (x - 1/2) < (8/(2+e))*(e/4)

-2e/(2+e) < (8x-4)/(2+e) < 2e/(2+e)

So now I'm stuck the other way because I can't reduce (8x-4)/(2+e) to (x-2).

There's more problems that seem to use the same kind of manipulation that I'm failing to find.

Thanks in advance.

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# Homework Help: Deleted neighborhoods, missing something about working with inequalities

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