Epsilon-Delta Definition of Limit (Proofs)

  • Thread starter Daniel Y.
  • Start date
  • #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.
 

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
25,832
251
Hi Daniel! :smile:
… 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
21
0
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
 

Related Threads on Epsilon-Delta Definition of Limit (Proofs)

  • Last Post
Replies
5
Views
3K
  • Last Post
Replies
20
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
4
Views
5K
  • Last Post
Replies
22
Views
8K
  • Last Post
Replies
15
Views
9K
  • Last Post
Replies
4
Views
9K
  • Last Post
Replies
8
Views
2K
Replies
13
Views
1K
  • Last Post
Replies
9
Views
26K
Top