Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Need Help with Formal Definition of Limits

  1. Oct 6, 2011 #1
    1. The problem statement, all variables and given/known data
    Limit a[itex]\underline{}n[/itex] as n→∞ = a. Find the limit a, and Determine N so that absolute value(a[itex]\underline{}n[/itex] - a) < [itex]\epsilon[/itex] for all n>N for the given value of [itex]\epsilon[/itex].

    The problem that I am working on is:

    a[itex]\underline{}n[/itex] = 1/n , [itex]\epsilon[/itex] = 0.01

    I'm sure this is very simple, as I am only two weeks into my university's basic calcuus class, but I am not nderstanding what to do. I have also tried going to tutoring and office hours, but my professor only confuses me more with his broken English.

    2. Relevant equations

    I am not sure what N is. I know that n is the nmber we are currently plugging in. I also know that a[itex]\underline{}n[/itex] is the whatever equation we are using (in this problem it is 1/n), and I know that [itex]\epsilon[/itex] is a margin above and below the limit.

    3. The attempt at a solution

    I saarted with the equation:
    absolute value((a[itex]\underline{}n[/itex]) - a) <[itex]\epsilon[/itex]

    I then plugged in numbers to get:
    absolute value ((1/n)-0) < 0.01

    After dropping the absolute value (because the limit is zero, and I think I am only solving for positive[itex]\epsilon[/itex]), and isolating n, I proceeded to get:
    100 < n

    I do not know what to do from here. I am not sure what n>N means or how to solve for it. Thank you so much.
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Oct 6, 2011 #2
    N is an unknown quantity that you have to find. Its value depends upon ε. Perhaps thinking of it this way will help: We're playing a game. I give you a specific value for ε. You have to give me back a value for N such that any time that n > N, then 1/n < ε. So, I give you ε = 0.01. You have to find an N such that whenever n > N, then 1/n < 0.01. What value of N would work? You've already done most of the work. You just have to put it together. Hope this helps.
  4. Oct 6, 2011 #3
    Thank you for the reply, I really appreciate it.

    Please correct me if I'm wrong, but because n > 100, and because n > N, we could set N = 100. This means that 1/n < 1/N → 1/n < 1/100 → 1/n < ε. Is that seriously the answer, because if so, I want to slap myself in the face right now.
  5. Oct 6, 2011 #4


    User Avatar
    Science Advisor

    Yes, that is seriously the answer- you may now slap!

    Obviously, the limit is 0 so, essentially, you want |(1/n)- 0|< .01. Of course, 1/n- 0= 1/n and since n> 0 that is the same as 1/n< .01 so n> 100. Choose N to be any number greater than or equal to 100 and it follows that if n> 100, then 1/n= |1/n- 0|< 0.01.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook