Simple absolute value problem with inequalities

In summary, the problem is to find the value of d in terms of e, for all e>0 and d>0, such that |x-a|<d implies |\sqrt x - \sqrt a| < e, where f(x) = sqrt(x). After some algebraic manipulation, the solution is d = e^2 - 2(a - sqrt(xa)), which depends on x. However, if the goal is to prove the continuity of sqrt(x) for all positive values of x, then d does not have to be independent of x.
  • #1
complexhuman
22
0
"Simple" absolute value problem with inequalities

OK...Im totally stuck and could use some help :)
given...for all e>0, d>0...the following holds
|x-a|<d => |f(x) - f(a)| < e
where f(x) = sqrt(x)

how do I find d in terms of e?


Thanks in advance
 
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  • #2
|x - a| = |[f(x) - f(a)][f(x) + f(a)| < e.|f(x) + f(a)| < d
I don't think you can get simpler than that.
 
  • #3
I am going to assume you meant: for all e > 0 there is some d > 0 such that |[itex]x-a[/itex]|< d implies |[itex]\sqrt x - \sqrt a[/itex]| < e.

([itex]\sqrt x - \sqrt a[/itex])2 < e2

[itex]x + a - 2\sqrt{x a}[/itex] < e2

[itex]x - a + 2(a-\sqrt{x a})[/itex] < e2

[itex]x - a[/itex] < e2 - [itex]2(a-\sqrt{x a})[/itex]

If [itex]x - a[/itex] > 0 then d = e2 - [itex]2(a-\sqrt{x a})[/itex]

(So d depends on x, and I guess that's okay.)

I need to think about the case where [itex]x - a[/itex] < 0.
 
  • #4
It is simplest to note that:
[tex]|\sqrt{x}-\sqrt{a}|=\frac{|x-a|}{\sqrt{x}+\sqrt{a}}[/tex]
and proceed from there.
 
  • #5
well...I end up with something like [tex]|x-a|<d => |x-a|=e|\sqrt{x}+\sqrt{a}|[/tex]...And that's where I am stuck on :(

yah...d has to be independent of x...its one of those proving limit typa thing. I am just allowed to assume a = 4 first
 
  • #6
If you are attempting to prove that [itex]\sqrt{x}[/itex] is continuous for all positive values of x, then d does not have to be independent of d. That's only true for uniform continuity.

If you have [itex]|x-a| |x-a|=e|\sqrt{x}+\sqrt{a}|[/itex] and x is "sufficiently close to a", say, |x-a|< 1/2, so that a- 1/2< x< a+ 1/2, what can you say about [itex]\sqrt{x}+ \sqrt{a}[/itex]?
 
  • #7
complexhuman said:
well...I end up with something like [tex]|x-a|<d => |x-a|=e|\sqrt{x}+\sqrt{a}|[/tex]...And that's where I am stuck on :(

yah...d has to be independent of x...its one of those proving limit typa thing. I am just allowed to assume a = 4 first
hehe...how did you get [tex]|x-a|<d => |x-a|=e|\sqrt{x}+\sqrt{a}|[/tex]?
I think it should be [tex]|x-a|<d => |x-a|<e|\sqrt{x}+\sqrt{a}|[/tex]
 

1. What is an absolute value?

An absolute value is the numerical value of a number without considering whether it is positive or negative. It is represented by two vertical bars surrounding the number.

2. How do you solve a simple absolute value problem with inequalities?

To solve a simple absolute value problem with inequalities, first isolate the absolute value expression on one side of the equation. Then, write two separate equations without the absolute value bars, one with a positive sign and one with a negative sign. Solve both equations separately and the solutions will represent the two possible values for the variable.

3. What is the difference between an absolute value equation and an absolute value inequality?

An absolute value equation is an equation that contains an absolute value expression, while an absolute value inequality is an inequality that contains an absolute value expression. The main difference is that when solving an absolute value equation, you will get one or two solutions, whereas when solving an absolute value inequality, you will get a range of solutions.

4. Can absolute value be negative?

No, absolute value cannot be negative. The absolute value of a number is always positive or zero.

5. What are some real-life applications of absolute value inequalities?

Absolute value inequalities are commonly used in fields such as economics, physics, and engineering to model real-life situations. For example, they can be used to represent constraints in optimization problems or to determine the range of values for a certain variable in a given situation.

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