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Math Help - Pointwise limits

  1. #1
    Senior Member
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    Pointwise limits

    I'm trying to understand how to do pointwise limits
    a)f_n(x)=nx/(1+nx^2)
    b)g_n(x)=(nx+sin(nx))/2n
    lim[1/2+sin(nx)/2n)]
    c)h_n(x)=x/(1+x^n)
    d)f_n(x)=1 if |x|>=1/n
    =n|x| if |x|<1/n
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  2. #2
    Member Abu-Khalil's Avatar
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    When you study pointwise limits, x is a constant.

    For example

    \displaystyle\lim_{n\to\infty}f_n(x)=\lim_{n\to\in  fty}\frac{nx}{1+nx^2}=\lim_{n\to\infty}\frac{x}{\f  rac 1n+x^2}=\frac{x}{x^2}=\frac 1x.

    Also, you can conclude the convergence isn't uniform on \mathbb R 'cause each f_n is continuous but the limit is not continuous.
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