Epsilon-Delta Definition of Limit (Proofs)

In summary, the conversation discusses the definition of a limit and the difficulty of proving limits for higher order functions. The speaker also asks for help in understanding how to make the connection between epsilon and delta in these proofs. They are directed to a helpful example for clarification.
  • #1
Daniel Y.
In my self-study Calculus book I finished with the 'intuitive' definition of the limit and the text directed me to the 'formal' definition of the limit. After reading the section covering it a few times I think I comprehended the details of the rigorous rules dictating it - but obviously not well enough.

The problem is I'm having trouble proving limits for functions of the second order (I find the limit and prove it is so). For instance, the limit of 3x-1 as x approaches 2 is fairly trivial, but, say, the limit of (x^2) - 3 as x approaches 2 confuses me. I try to figure out a value to let epsilon = delta be, but get to the point where epsilon > |x-2||x+2|, and don't know how to 'make the connection' between it and delta > |x-2|, letting (epsilon)/|x+2| = delta doesn't seem right. If you could help me get my head around proving limits for these higher order functions, I'd really appreciate it. Thanks.
 
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  • #2
Hi Daniel! :smile:
Daniel Y. said:
… letting (epsilon)/|x+2| = delta doesn't seem right.

That's because δ must be a function of ε (and not of x at all).

Given ε, you need to find a δ such that if |x-2| < δ, then |(x² - 3) - 1| < ε.

So just jiggle about with this until you find a function of ε which works (though it won't be a nice linear one). :smile:
 
  • #3
Daniel Y. said:
In my self-study Calculus book I finished with the 'intuitive' definition of the limit and the text directed me to the 'formal' definition of the limit. After reading the section covering it a few times I think I comprehended the details of the rigorous rules dictating it - but obviously not well enough.

The problem is I'm having trouble proving limits for functions of the second order (I find the limit and prove it is so). For instance, the limit of 3x-1 as x approaches 2 is fairly trivial, but, say, the limit of (x^2) - 3 as x approaches 2 confuses me. I try to figure out a value to let epsilon = delta be, but get to the point where epsilon > |x-2||x+2|, and don't know how to 'make the connection' between it and delta > |x-2|, letting (epsilon)/|x+2| = delta doesn't seem right. If you could help me get my head around proving limits for these higher order functions, I'd really appreciate it. Thanks.

I think this will answer your question , look at example 3

http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfLimit.aspx
 

1. What is the Epsilon-Delta Definition of Limit?

The Epsilon-Delta Definition of Limit is a mathematical concept used to formally define the limit of a function. It states that a function f(x) has a limit L as x approaches a point c if, for any positive value of epsilon (ε), there exists a positive value of delta (δ) such that if the distance between x and c is less than delta, then the distance between f(x) and L is less than epsilon.

2. Why is the Epsilon-Delta Definition of Limit important?

The Epsilon-Delta Definition of Limit is important because it provides a rigorous and precise way to determine the limit of a function. It allows mathematicians to prove the existence of a limit and to determine the value of the limit without relying on graphical or intuitive arguments.

3. How do you use the Epsilon-Delta Definition of Limit to prove the limit of a function?

To prove the limit of a function using the Epsilon-Delta Definition, you must first choose a value for epsilon (ε) and then use algebraic manipulations to find a value for delta (δ) that satisfies the definition. This usually involves setting an upper bound for the absolute value of f(x) - L and then finding a corresponding upper bound for the absolute value of x - c.

4. What are the common mistakes when using the Epsilon-Delta Definition of Limit?

One common mistake when using the Epsilon-Delta Definition of Limit is to fix the value of delta (δ) and try to find a corresponding value for epsilon (ε). This can lead to incorrect results, as the definition states that the value of delta must depend on the value of epsilon. Another mistake is to assume that the value of delta must be the same for all values of epsilon, when in fact it can vary depending on the function and the point c being approached.

5. How does the Epsilon-Delta Definition of Limit relate to the concept of continuity?

The Epsilon-Delta Definition of Limit is closely related to the concept of continuity. A function f(x) is continuous at a point c if and only if the limit of f(x) as x approaches c exists and is equal to f(c). This means that a function is continuous if and only if the Epsilon-Delta Definition of Limit is satisfied for all values of epsilon (ε) and delta (δ).

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