What is the Optimal Delta for a Given Epsilon and Limit?

In summary, to find the greatest c such that every δ between zero and c is good, we solve the equation (x+2)/(x-3) = -2/3 ± 0.01 and choose the smaller of the two solutions. This ensures that the statement "if |x| < δ, then |f(x)| < ε" is true for all δ between zero and c.
  • #1
ƒ(x)
328
0
Given the limit of [tex]\frac{x^2+2x}{x^2-3x}[/tex] as x approaches 0 equals [tex]\frac{-2}{3}[/tex] and that ε = .01, find the greatest c such that every δ between zero and c is good. Give an exact answer.


0 < |x-0| < δ
0 < |[tex]\frac{x^2+2x}{|x^2-3x}[/tex] + [tex]\frac{2}{3|}[/tex]| < ε


|[tex]\frac{x(x+2)}{|x(x-3)}[/tex] + [tex]\frac{2}{3|}[/tex]| = .01
|[tex]\frac{x+2}{|x-3|}[/tex]| = [tex]\frac{-197}{300}[/tex]
300(x+2) = -197(x-3)
300x + 600 = -197x + 591
497x = -9
x = [tex]\frac{-9}{497}[/tex]

But, my book got [tex]\frac{9}{503}[/tex]
 
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  • #2
ƒ(x) said:
x = [tex]\frac{-9}{497}[/tex]

But, my book got [tex]\frac{9}{503}[/tex]

Hi ƒ(x)! :smile:

The ε has to be valid on both sides of -2/3 (and positive) …

that gives you 9/497 and 9/503, and 9/503 is smaller. :wink:
 
  • #3
Ok, but why do I pick the smaller one? If the larger delta works then every delta smaller than that will also work, right?

Also, does it matter if its negative since there is an absolute value sign in the inequality?
 
  • #4
Hi ƒ(x)! :smile:

(just got up :zzz: …)
ƒ(x) said:
Ok, but why do I pick the smaller one? If the larger delta works then every delta smaller than that will also work, right?

Because if it works for -δ1 < x < δ2, and δ1 > δ > δ2,

then it does not work for x = -δ, does it? :wink:
Also, does it matter if its negative since there is an absolute value sign in the inequality?

The definition says |x| < δ, so δ must be positive.
 
  • #5
Ok, it's beginning to click. Could you give me a walk through of how you would do this problem?
 
  • #6
ƒ(x) said:
Ok, it's beginning to click. Could you give me a walk through of how you would do this problem?

I'd solve (x+2)/(x-3) = -2/3 ± 0.01 (isn't that what you did?), and take the smaller of those two |x|s.
 
  • #7
Ok. Can you explain again why you take the smaller of the two values?
 
  • #8
You've found f(-δ1) = f(δ2) = ± ε.

Now you need a δ such that if |x| < δ, then |f(x)| < ε.

If δ1 > δ2, and if we choose δ = δ1, then the statement "if |x| < δ1, then |f(x)| < ε" isn't true, because although f(-δ1) = ε, f(δ1) > ε. :smile:
 

1. What is the definition of epsilon delta limits?

Epsilon delta limits, also known as the precise definition of a limit, is a mathematical concept used to formally define the behavior of a function as its input value approaches a certain value. It involves the use of two variables, epsilon and delta, to represent a range of values in which the output of the function will fall within for a given input value.

2. How do you prove a limit using epsilon delta?

To prove a limit using epsilon delta, one must show that for any positive value of epsilon, there exists a corresponding positive value of delta such that the output of the function falls within the range of epsilon for all input values within the range of delta. This is typically done by manipulating the definition of the limit and using algebraic techniques to solve for an appropriate value of delta.

3. What is the purpose of using epsilon delta limits?

The purpose of using epsilon delta limits is to provide a rigorous and precise definition of a limit, which allows for a more exact understanding of the behavior of a function as its input value approaches a certain value. It also allows for the evaluation of limits at points where the function may not be defined or where there may be discontinuities.

4. Can epsilon delta limits be used for all types of functions?

Yes, epsilon delta limits can be used for all types of functions, including polynomial, trigonometric, and exponential functions. However, the process of proving a limit using epsilon delta may vary depending on the type of function and the specific value of the limit being evaluated.

5. How does the concept of epsilon delta limits relate to calculus?

Epsilon delta limits are a fundamental concept in calculus and are used extensively in the study of limits, derivatives, and integrals. They provide a rigorous foundation for the calculus operations of differentiation and integration, and are essential for understanding the behavior of functions in a precise and mathematical way.

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