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Math Help - Limits

  1. #1
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    Limits

    Hi I am having problems with a couple of questions on my current assignment. I assume both questions are relating to the delta/epsilon proof of limits, which was given alongside the assignment.

    Lets f1(x) = x, f2(x) = 2x, ..., fn(x) = nx. Then lim[x->0] fn(x)=0 for each n. Does there exist a positive number delta such that for every positive integer n we have absolute(fn(x)-0) <1 whenever 0< absolute(x-0) < delta


    Let f(x) = 1 if x is rational, 0 if x is irrational. Show that lim[x->0] f(x) does not exist.
    Last edited by Webby; April 25th 2010 at 06:24 AM.
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  2. #2
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    Quote Originally Posted by Webby View Post
    Hi I am having problems with a couple of questions on my current assignment. I assume both questions are relating to the delta/epsilon proof of limits, which was given alongside the assignment.

    Lets f1(x) = x, f2(x) = 2x, ..., fn(x) = nx. Then lim[x->0] fn(x)=0 for each n. Does there exist a positive number delta such that for every positive integer n we have absolute(fn(x)-0) <1 whenever 0< absolute(x-0) < delta


    Let f(x) = 1 if x is rational, 0 if x is irrational. Show that lim[x->0] f(x) does not exist.

    1. No

    For any  \delta > 0 , we can find a natural number N such that  x^* = 1/N < \delta but  f_N(x^*) = N*\frac{1}{N} = 1

    2. Hint: look at  \epsilon < 1 and use the fact that rational numbers are dense on any interval to show that the definition of continuity will not hold.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Webby View Post


    Let f(x) = 1 if x is rational, 0 if x is irrational. Show that lim[x->0] f(x) does not exist.
    A very useful nugget of information to have locked up in your noodle is that since \mathbb{Q},\mathbb{R}-\mathbb{Q} are dense in \mathbb{R} for each x\in\mathbb{R} there are sequences \{q_n\},\{i_n\} in each such that \lim\text{ }q_n=x=\lim\text{ }i_n.
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