# Find an upper bound to the limit of a function

1. Oct 12, 2009

### zeion

1. The problem statement, all variables and given/known data

Find an upper bound M for f(x) = |x-2 / x+(1/2)| if |x+1| < 1/4

2. Relevant equations

3. The attempt at a solution

I'm confused about this |x+1| < 1/4. Does this mean that |x-1| < 1/4?

|x-2/x+(1/2)| = x-2/(2x+1)/2 = 2(x-2)/(2x+1) = 2x - 4/2x + 1 = x-2/x+(1/2) < M

Given -1/4 < |x+1| < 1/4
-5/4 < x < -3/4
-13/4 < x - 2 < -11/4

Also
-3/4 < x + (1/2) < -1/4
then
-4 < 1/ x + (1/2) < -4/3
3 < x - 2 / x + (1/2) < 1/3 ???

2. Oct 12, 2009

### LCKurtz

"I'm confused about this |x+1| < 1/4. Does this mean that |x-1| < 1/4?"

No. Why would it mean that? You have an (x + 1/2) in the denominator. (At least I suppose you do; it should have parentheses around it.) So you don't want x to get too close to -1/2. Doesn't |x+1| < 1/4 help you with that?

3. Oct 12, 2009

### zeion

If |x+1| < 1/4
-1/4 < x+1 < 1/4
-5/4 < x < -3/4
x cannot = -1/2

What does "upper bound M" mean?

4. Oct 12, 2009

### Staff: Mentor

A number M such that f(x) <= M for all x in some set. For example, M = 1 is an upper bound for g(x) = sin(x), for all real numbers x. Also, 2 is an upper bound, as is 1.7. 1 is the least upper bound.