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Continuity of exp(x)

  1. Feb 20, 2012 #1
    1. The problem statement, all variables and given/known data
    use delta, epsilon to prove that e^x is continuous at c = 0

    2. Relevant equations

    (a) for y>0, lim_n-> inf, y^(1/n) = 1
    (b) for x < y, exp(x) < exp(y)

    3. The attempt at a solution

    im not sure how to approach this problem.
    i have,
    |exp(x) - exp(0)|= |exp(x) - 1|
    so then exp(x) < 1 + ε
    for δ > 0,
    exp(δ) < 1 + ε

    so then, i would set δ=ln(1+ε) for the proof?

    also im not sure how to use relevant eqn (a) to help with the problem. some insight would be appreciated. thank you.
  2. jcsd
  3. Feb 20, 2012 #2
    You are not supposed to use log; you are supposed to use the two conditions (a) and (b); first you realize that the limit in (a) is fairly close to your limit, if you set y=e, then you can use that limit to come up with a delta which is a function of the N(epsilon) of that limit.
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