Calculating limits of function












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On practical classes at university we calculated some limits of functions of the following type:
$$lim_{xto 0}f(x)^{k(x)}$$
The key idea was that we know that $exists lim_{xto 0}f(x) = p in mathbb{R}$ (usually $p = e$). So we put this result in our initial expression:
$$lim_{xto 0}f(x)^{k(x)} = lim_{xto 0}p^{k(x)}$$
How can this step be proved formally? Probably, to prove this should be used variable substitution, but in such case I have ho ideas how to deal with $k(x)$.



upd: Ok, this cannot be proved in general, but what if $f(x) =(1+x)^{frac{1}{x}}$?






real-analysis limits







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    On practical classes at university we calculated some limits of functions of the following type:
    $$lim_{xto 0}f(x)^{k(x)}$$
    The key idea was that we know that $exists lim_{xto 0}f(x) = p in mathbb{R}$ (usually $p = e$). So we put this result in our initial expression:
    $$lim_{xto 0}f(x)^{k(x)} = lim_{xto 0}p^{k(x)}$$
    How can this step be proved formally? Probably, to prove this should be used variable substitution, but in such case I have ho ideas how to deal with $k(x)$.



    upd: Ok, this cannot be proved in general, but what if $f(x) =(1+x)^{frac{1}{x}}$?






    real-analysis limits







    share|cite|improve this question



























      1












      1








      1







      On practical classes at university we calculated some limits of functions of the following type:
      $$lim_{xto 0}f(x)^{k(x)}$$
      The key idea was that we know that $exists lim_{xto 0}f(x) = p in mathbb{R}$ (usually $p = e$). So we put this result in our initial expression:
      $$lim_{xto 0}f(x)^{k(x)} = lim_{xto 0}p^{k(x)}$$
      How can this step be proved formally? Probably, to prove this should be used variable substitution, but in such case I have ho ideas how to deal with $k(x)$.



      upd: Ok, this cannot be proved in general, but what if $f(x) =(1+x)^{frac{1}{x}}$?






      real-analysis limits







      share|cite|improve this question















      On practical classes at university we calculated some limits of functions of the following type:
      $$lim_{xto 0}f(x)^{k(x)}$$
      The key idea was that we know that $exists lim_{xto 0}f(x) = p in mathbb{R}$ (usually $p = e$). So we put this result in our initial expression:
      $$lim_{xto 0}f(x)^{k(x)} = lim_{xto 0}p^{k(x)}$$
      How can this step be proved formally? Probably, to prove this should be used variable substitution, but in such case I have ho ideas how to deal with $k(x)$.



      upd: Ok, this cannot be proved in general, but what if $f(x) =(1+x)^{frac{1}{x}}$?






      real-analysis limits




      real-analysis limits






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      share|cite|improve this question













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      edited Nov 26 at 18:23

























      asked Nov 26 at 16:01









      Yan Kardziyaka

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          This can't be proven formally, because it isn't true in general. Take $f(x) = 1 - x$, and $k(x) = frac{1}{x}$. Then $p = 1$, and so $lim_{xto 0}p^{k(x)} = 1$, but $lim_{xto 0}f(x)^{k(x)} = frac{1}{e}$.






          share|cite|improve this answer





















          • thanks. Can you find counterexample for $f(x) = (1 + x)^frac{1}{x}$ ?
            – Yan Kardziyaka
            Nov 26 at 16:22













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          This can't be proven formally, because it isn't true in general. Take $f(x) = 1 - x$, and $k(x) = frac{1}{x}$. Then $p = 1$, and so $lim_{xto 0}p^{k(x)} = 1$, but $lim_{xto 0}f(x)^{k(x)} = frac{1}{e}$.






          share|cite|improve this answer





















          • thanks. Can you find counterexample for $f(x) = (1 + x)^frac{1}{x}$ ?
            – Yan Kardziyaka
            Nov 26 at 16:22


















          2














          This can't be proven formally, because it isn't true in general. Take $f(x) = 1 - x$, and $k(x) = frac{1}{x}$. Then $p = 1$, and so $lim_{xto 0}p^{k(x)} = 1$, but $lim_{xto 0}f(x)^{k(x)} = frac{1}{e}$.






          share|cite|improve this answer





















          • thanks. Can you find counterexample for $f(x) = (1 + x)^frac{1}{x}$ ?
            – Yan Kardziyaka
            Nov 26 at 16:22
















          2












          2








          2






          This can't be proven formally, because it isn't true in general. Take $f(x) = 1 - x$, and $k(x) = frac{1}{x}$. Then $p = 1$, and so $lim_{xto 0}p^{k(x)} = 1$, but $lim_{xto 0}f(x)^{k(x)} = frac{1}{e}$.






          share|cite|improve this answer












          This can't be proven formally, because it isn't true in general. Take $f(x) = 1 - x$, and $k(x) = frac{1}{x}$. Then $p = 1$, and so $lim_{xto 0}p^{k(x)} = 1$, but $lim_{xto 0}f(x)^{k(x)} = frac{1}{e}$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 26 at 16:06









          user3482749

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          2,411414












          • thanks. Can you find counterexample for $f(x) = (1 + x)^frac{1}{x}$ ?
            – Yan Kardziyaka
            Nov 26 at 16:22




















          • thanks. Can you find counterexample for $f(x) = (1 + x)^frac{1}{x}$ ?
            – Yan Kardziyaka
            Nov 26 at 16:22


















          thanks. Can you find counterexample for $f(x) = (1 + x)^frac{1}{x}$ ?
          – Yan Kardziyaka
          Nov 26 at 16:22






          thanks. Can you find counterexample for $f(x) = (1 + x)^frac{1}{x}$ ?
          – Yan Kardziyaka
          Nov 26 at 16:22




















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