Do limits respect composition of functions?

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The question arose when I was trying to prove the chain rule for Gateaux differentials. Here’s what I have.



$$
d(gcirc f)_p(v) = lim_tto 0 fracg(f(p+tv)) - g(f(p))t = lim_tto 0 fracg(f(p) + tphi_p(v)) - g(f(p))t
\
= lim_tto 0 fracg(f(p)) + tgamma_f(p)(phi_p(v)) -g(f(p))t = lim_tto 0 gamma_f(p)(phi_p(v))
$$



where
$$
phi_p(v) = fracf(p+tv) - f(p)t quadquad
gamma_f(p)(phi_p(v)) = fracg(f(p) + tphi_p(v)) - g(f(p))t
$$



So clearly
$$
lim_tto 0 phi_p = (df)_p text and lim_tto 0 gamma_f(p) = (dg)_f(p)
$$



But can they happen simultaneously? I.e. Is the limit of the composition equal to the composition of the limits?




limits multivariable-calculus




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    up vote
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    down vote

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    The question arose when I was trying to prove the chain rule for Gateaux differentials. Here’s what I have.



    $$
    d(gcirc f)_p(v) = lim_tto 0 fracg(f(p+tv)) - g(f(p))t = lim_tto 0 fracg(f(p) + tphi_p(v)) - g(f(p))t
    \
    = lim_tto 0 fracg(f(p)) + tgamma_f(p)(phi_p(v)) -g(f(p))t = lim_tto 0 gamma_f(p)(phi_p(v))
    $$



    where
    $$
    phi_p(v) = fracf(p+tv) - f(p)t quadquad
    gamma_f(p)(phi_p(v)) = fracg(f(p) + tphi_p(v)) - g(f(p))t
    $$



    So clearly
    $$
    lim_tto 0 phi_p = (df)_p text and lim_tto 0 gamma_f(p) = (dg)_f(p)
    $$



    But can they happen simultaneously? I.e. Is the limit of the composition equal to the composition of the limits?




    limits multivariable-calculus




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      The question arose when I was trying to prove the chain rule for Gateaux differentials. Here’s what I have.



      $$
      d(gcirc f)_p(v) = lim_tto 0 fracg(f(p+tv)) - g(f(p))t = lim_tto 0 fracg(f(p) + tphi_p(v)) - g(f(p))t
      \
      = lim_tto 0 fracg(f(p)) + tgamma_f(p)(phi_p(v)) -g(f(p))t = lim_tto 0 gamma_f(p)(phi_p(v))
      $$



      where
      $$
      phi_p(v) = fracf(p+tv) - f(p)t quadquad
      gamma_f(p)(phi_p(v)) = fracg(f(p) + tphi_p(v)) - g(f(p))t
      $$



      So clearly
      $$
      lim_tto 0 phi_p = (df)_p text and lim_tto 0 gamma_f(p) = (dg)_f(p)
      $$



      But can they happen simultaneously? I.e. Is the limit of the composition equal to the composition of the limits?




      limits multivariable-calculus




      share|cite|improve this question











      The question arose when I was trying to prove the chain rule for Gateaux differentials. Here’s what I have.



      $$
      d(gcirc f)_p(v) = lim_tto 0 fracg(f(p+tv)) - g(f(p))t = lim_tto 0 fracg(f(p) + tphi_p(v)) - g(f(p))t
      \
      = lim_tto 0 fracg(f(p)) + tgamma_f(p)(phi_p(v)) -g(f(p))t = lim_tto 0 gamma_f(p)(phi_p(v))
      $$



      where
      $$
      phi_p(v) = fracf(p+tv) - f(p)t quadquad
      gamma_f(p)(phi_p(v)) = fracg(f(p) + tphi_p(v)) - g(f(p))t
      $$



      So clearly
      $$
      lim_tto 0 phi_p = (df)_p text and lim_tto 0 gamma_f(p) = (dg)_f(p)
      $$



      But can they happen simultaneously? I.e. Is the limit of the composition equal to the composition of the limits?





      limits multivariable-calculus





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









      asked Jul 21 at 2:32









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