Question 425330: I didn't see a Calculus section so I'm posting my question here. I need to know what I'm doing wrong with this definition of a limit. Please help. Thanks!
The question: What is the formal definition of a limit? Given an example (not using numbers) that is relevant to prove the definition.
I have submitted the following answer and it is incorrect somewhere:
The formal definition of a limit in regards to the limits of a function would be described as the limit of f(x) as x approaches a is equal to L. The formula for this definition looks like this: lim f(x)=L.
In the formula, L represents the limit of x, f(x) is the function notation, x represents a positive number or the absolute value of a negative number, and a is the number associated with x as it approaches its limit. Another way to define this is that the limit of f(x) as x approaches a is equal to L if for every number (represented by the symbol epsilon) larger than zero there is a corresponding number (represented by the symbol delta). The absolute value is less than epsilon when the absolute value of x – a is greater than zero and when the absolute value of x – a is less than delta.
This limit means that you can force f(x) to be as close as you want to a number L by choosing an x value close enough to a. The area between f(x) and L is delta and the difference between x and a is called epsilon. This can be done for any number as long as the values that are chosen for delta and epsilon are positive. The objective is to find a δ > 0 such that |f(x) - a| < δ when 0 < |x - a| < ε.
|f(x) - a| < δ is the same as |x - a| < δ. So choose δ = ε. For example, you need f(x) to be within delta units of L. To make this happen, you need to choose numbers for x that are within epsilon units of a. Delta and epsilon must be positive. Let’s suppose, for example, that f(x) = x. The limit as x approaches -1 is -1. Finding the limits of a function can be done using any arbitrary number so long as the values of delta and epsilon are positive. It is important to note that the value for delta is equal to the value of epsilon.
This is what I have submitted. They want an example of how delta can be written in terms of epsilon. There are apparently incorrect statements also in the first paragraph. Could someone please look at this for me? I only have one more try to get this essay correct. Thanks so much for all your help!
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Please check the following site.
If you want other explanations of epsilon-delta limits
please use Google to search the web. There are plenty.
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http://www.scottsarra.org/applets/calculus/EpsilonDelta.html
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Cheers,
Stan H.
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