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Math Help - Question about limits

  1. #1
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    Question about limits

    In part (a) of my questions I'm asked to show \lim_{n \rightarrow{\infty}}[\frac{1}{n-1}]^\frac{1}{n}  = 1

    I have done this, and part (b) asks me to consider a sequence of functions

    f_n : [0,1] \to \mathbb{R} : x \to \frac{x}{1 + x^n},

    then for each fixed n \in \mathbb{N} determine the point \varepsilon(n) \in [0,1] at which the function f_n attains its maximum value and hence, calculate

    \lim_{n \rightarrow{\infty}}f_n(\varepsilon(n))

    I'm told I may use part a. I'm not sure if \varepsilon(n) is meant to represent x values in which case the max would be achieved (as I can see) at x = 1, but I can't see the relation with part (a) by doing that.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by pineapple89 View Post
    In part (a) of my questions I'm asked to show \lim_{n \rightarrow{\infty}}[\frac{1}{n-1}]^\frac{1}{n}  = 1

    I have done this, and part (b) asks me to consider a sequence of functions

    f_n : [0,1] \to \mathbb{R} : x \to \frac{x}{1 + x^n},

    then for each fixed n \in \mathbb{N} determine the point \varepsilon(n) \in [0,1] at which the function f_n attains its maximum value and hence, calculate

    \lim_{n \rightarrow{\infty}}f_n(\varepsilon(n))

    I'm told I may use part a. I'm not sure if \varepsilon(n) is meant to represent x values in which case the max would be achieved (as I can see) at x = 1, but I can't see the relation with part (a) by doing that.
    Let's solve  f_n'(x)=0 \implies \frac{1-(n-1)x^n}{(x^n+1)^2}=0 \implies 1-(n-1)x^n=0 \implies x=\left(\frac{1}{n-1}\right)^\frac{1}{n} .

    Hence  f_n(x) has a critical point at  x=\left(\frac{1}{n-1}\right)^\frac{1}{n} . Now verify that this is indeed a maximum.
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  3. #3
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    Aah, I see, I was thinking in terms of looking for the supremum like when establishing uniform convergence. Didn't think of maximum like that. Silly me Thanks!
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