Mastering Epsilon-Delta Proofs: How to Find Delta When x Approaches a Value 'a'?

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In summary, the conversation is about the difficulty of understanding and using delta/epsilon proofs in limits where x approaches a value 'a'. The person is seeking help in finding the usual procedure for finding delta and mentions using an 'a' term and an 'x' term. Another person responds with an example using the function 1/x and explains how to manipulate the terms to make the bottom fraction as large as possible. They also mention the importance of playing around with the terms and suggest using the textbook Thomas and Finney for practice.
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phos
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Hello,

I have been looking around the net recently trying to teach myself how to do delta/epsilon proofs. unfortunately, my prof sucks :frown: . Anyway, I think I understand how to get delta in terms of epsilon for limits where x approaches a number, but I'm having difficulty finding what to do when x approaches a value 'a'.

eg. prove: lim of x approaches a ( 1/x^3) = (1/a^3)


I don't really need a solution, although it would be welcome. I'm just interested in what the usual procedure is for finding delta as I'm always stuck with an 'a' term and an 'x' term that I'm not sure what to do with or how to get rid of. Any help is greatly appreciated.
 
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  • #2
i usually write x as a+h, whgere h is basically delta, and then subtract f(x) from f(a) to see how big it is. then i try to make it elss than epsilon.

for example if i have f(x) = 1/x, i call it 1/(a+h) and look at 1/(a+h) - 1/a.

assume a >0.

that equals (a - [a+h])/[a(a+h)] = -h/[a(a+h)]. now i want this to be small, so i want the bottom to be large, and i need to get a handle on h. If h were too close to -a the bottom would get too small so I choose h first of all to be less than a/2. Then a+h is between a/2 and 3a/2. so the bottom is between a^2/2 and 3a^2/2.

remember i want the bottom to be large, so i know the bottom is at least as large as a^2/2, whenever h is less than a/2.

now that emans the fraction h/[a(a+h)], is at least as small as h/[a^2/2] = 2h/a^2.

so if i can also make 2h/a^2 less than epsilon i am home. well that is easy to solve for.

i need h less than a^2/2 times epsilon, and also less than a/2.

i hated these fraction ones as a young student because you have to just play around with them likwe this, but see if you can check this out and then if it works, generalize it to your case.
 
  • #3
Saying your prof sucks isn't going to get any sympathy since most of the people here who answer questions are teachers who probablyu doubt your ability to make an unbiased judgement about someone trying to teach you a difficult subject that you don't yet understand.

As for this question, or many analysis questions, finding the exact eps-delta stuff is completely unnecessary: you know x tends to a as, erm, x tends to a, and that if x and y tend to a and b resp that xy tends to ab, and that if a is not zero 1/x tends to 1/a, hence result.
 
  • #4
Do you have Thomas and Finney? It has a nice introduction to epsilon-delta and the formal definition of limit. You should be able to get a good deal of practice from that book as well.

Cheers
Vivek
 

FAQ: Mastering Epsilon-Delta Proofs: How to Find Delta When x Approaches a Value 'a'?

What is an epsilon-delta proof?

An epsilon-delta proof is a rigorous mathematical technique used to prove the limit of a function. It involves choosing a small value, epsilon, and then finding a corresponding value, delta, such that if the input to the function is within delta of the limit, then the output will be within epsilon of the limit.

Why do we use epsilon-delta proofs?

Epsilon-delta proofs are used to provide a rigorous and formal way of proving limits. They allow us to make precise statements about the behavior of a function near a particular point, and provide a solid foundation for understanding the concept of a limit.

How do you construct an epsilon-delta proof?

To construct an epsilon-delta proof, you first start by writing down the definition of the limit. Then, you manipulate the expression to find a suitable expression for delta in terms of epsilon. Finally, you choose a specific value for delta that satisfies the conditions of the definition.

Can you give an example of an epsilon-delta proof?

One example of an epsilon-delta proof is proving the limit of the function f(x) = x^2 at x = 3. We start by writing down the definition of the limit: for any epsilon > 0, there exists a delta > 0 such that |x^2 - 9| < epsilon whenever 0 < |x - 3| < delta. Then, by manipulating the expression, we can choose delta = sqrt(epsilon + 9) - 3 to satisfy the conditions of the definition.

What are some common mistakes to avoid in an epsilon-delta proof?

Some common mistakes to avoid in an epsilon-delta proof include using the wrong definition of the limit, not manipulating the expression correctly, or not choosing a suitable value for delta. It is important to carefully follow the steps of the proof and to make sure that all conditions are satisfied in order to arrive at a correct and valid proof.

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