Epsilon-Delta Definition of Limit (Proofs)

[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.

 

[tiny-tim]

Hi Daniel!

… 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).

 

[lion0001]

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