Evaluate limit $a^xx^a$ where 0<a<1











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I want to evaluate the limit:$$lim_{x to infty}{x^aa^x} $$ where $0<a<1$



I put in Wolfram Alpha and I get the limit is zero



We see if the base is $0<a<1$ then :
$$lim_{x to infty}{a^x} =0$$



I tried to solve the limit with a teorem that says : $$ lim_{xto +infty} f(x) = 0$$ and $$g(x)$$ is a bounded .
Then $$lim_{xto +infty}f(x)g(x)=0$$



But $$lim_{x to infty}{x^a} =infty$$



and the product of $infty$ and $0$ is undefined










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  • Are you allowed to use L'Hôpital's rule?
    – Fimpellizieri
    Nov 15 at 4:57










  • @Fimpellizieri yes I allowed to use L'Hôpital rule
    – her
    Nov 15 at 5:02















up vote
0
down vote

favorite












I want to evaluate the limit:$$lim_{x to infty}{x^aa^x} $$ where $0<a<1$



I put in Wolfram Alpha and I get the limit is zero



We see if the base is $0<a<1$ then :
$$lim_{x to infty}{a^x} =0$$



I tried to solve the limit with a teorem that says : $$ lim_{xto +infty} f(x) = 0$$ and $$g(x)$$ is a bounded .
Then $$lim_{xto +infty}f(x)g(x)=0$$



But $$lim_{x to infty}{x^a} =infty$$



and the product of $infty$ and $0$ is undefined










share|cite|improve this question






















  • Are you allowed to use L'Hôpital's rule?
    – Fimpellizieri
    Nov 15 at 4:57










  • @Fimpellizieri yes I allowed to use L'Hôpital rule
    – her
    Nov 15 at 5:02













up vote
0
down vote

favorite









up vote
0
down vote

favorite











I want to evaluate the limit:$$lim_{x to infty}{x^aa^x} $$ where $0<a<1$



I put in Wolfram Alpha and I get the limit is zero



We see if the base is $0<a<1$ then :
$$lim_{x to infty}{a^x} =0$$



I tried to solve the limit with a teorem that says : $$ lim_{xto +infty} f(x) = 0$$ and $$g(x)$$ is a bounded .
Then $$lim_{xto +infty}f(x)g(x)=0$$



But $$lim_{x to infty}{x^a} =infty$$



and the product of $infty$ and $0$ is undefined










share|cite|improve this question













I want to evaluate the limit:$$lim_{x to infty}{x^aa^x} $$ where $0<a<1$



I put in Wolfram Alpha and I get the limit is zero



We see if the base is $0<a<1$ then :
$$lim_{x to infty}{a^x} =0$$



I tried to solve the limit with a teorem that says : $$ lim_{xto +infty} f(x) = 0$$ and $$g(x)$$ is a bounded .
Then $$lim_{xto +infty}f(x)g(x)=0$$



But $$lim_{x to infty}{x^a} =infty$$



and the product of $infty$ and $0$ is undefined







calculus limits






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share|cite|improve this question











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asked Nov 15 at 4:53









her

489




489












  • Are you allowed to use L'Hôpital's rule?
    – Fimpellizieri
    Nov 15 at 4:57










  • @Fimpellizieri yes I allowed to use L'Hôpital rule
    – her
    Nov 15 at 5:02


















  • Are you allowed to use L'Hôpital's rule?
    – Fimpellizieri
    Nov 15 at 4:57










  • @Fimpellizieri yes I allowed to use L'Hôpital rule
    – her
    Nov 15 at 5:02
















Are you allowed to use L'Hôpital's rule?
– Fimpellizieri
Nov 15 at 4:57




Are you allowed to use L'Hôpital's rule?
– Fimpellizieri
Nov 15 at 4:57












@Fimpellizieri yes I allowed to use L'Hôpital rule
– her
Nov 15 at 5:02




@Fimpellizieri yes I allowed to use L'Hôpital rule
– her
Nov 15 at 5:02










3 Answers
3






active

oldest

votes

















up vote
2
down vote



accepted










Here's a general result.




Theorem (Domination of exponentials and polynomials):
Let $a, b>0$.
Then
$$
lim_{xtoinfty}frac{x^a}{b^x} =
left{
begin{array}{cc}
+infty&text{if $,bleqslant 1$}\
0&text{if $,b> 1$}\
end{array}
right.
$$




Proof:
If $bleqslant 1$ then $b^xleqslant 1$ for all $xgeqslant 1$ and hence for $xgeqslant 1$ we have
$$
frac{x^a}{b^x} geqslant x^a stackrel{xtoinfty}{xrightarrow{hspace{1cm}}}+infty.
$$

If $b>1$ then $b^x to+infty$ as $xtoinfty$, so we may apply L'Hôpital's rule.
Observing that $b^x = expbig(xlog (b)big)$, we'll obtain that
$$
lim_{xtoinfty}frac{x^a}{b^x} =
lim_{xtoinfty} frac{a}{log(b)}frac{x^{a-1}}{b^x}
$$

Notice that up to the positive constant $a/log (b)$, we arrive at the same limit we began with, except the initial value of the exponent $a$ in the polynomial term has decreased by $1$.
We can successively apply L'Hôpital's rule until the exponent becomes non-positive, at which point the limit is easily seen to be $0$. $square$





For your particular question, we can apply the general result with $b=frac1a > 1$.






share|cite|improve this answer






























    up vote
    0
    down vote













    Let $x=1+t$ then Bernoulli's inequality shows
    $$lim_{x to infty}{x^aa^x}=lim_{t to infty}{(1+t)^aa^{1+t}}>lim_{t to infty}{(at)a^{1+t}}=a^2lim_{t to infty}{ta^{t}}=0$$
    the last one holds by L'Hospital.






    share|cite|improve this answer




























      up vote
      0
      down vote













      Consider
      $$y=a^x ,x^a implies log(y)=a log(x)+x log(a)=x left(afrac{ log (x)}{x}+log (a)right)$$ Since $xto infty$, $frac{ log (x)}{x}$ is negligible for any $a$. So
      $$log(y) sim x log(a)$$ and, since $0 < a < 1$, $log(a) <0$. Then $log(y)to -infty$ and $y=e^{log(y)} to 0$






      share|cite|improve this answer





















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        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes








        up vote
        2
        down vote



        accepted










        Here's a general result.




        Theorem (Domination of exponentials and polynomials):
        Let $a, b>0$.
        Then
        $$
        lim_{xtoinfty}frac{x^a}{b^x} =
        left{
        begin{array}{cc}
        +infty&text{if $,bleqslant 1$}\
        0&text{if $,b> 1$}\
        end{array}
        right.
        $$




        Proof:
        If $bleqslant 1$ then $b^xleqslant 1$ for all $xgeqslant 1$ and hence for $xgeqslant 1$ we have
        $$
        frac{x^a}{b^x} geqslant x^a stackrel{xtoinfty}{xrightarrow{hspace{1cm}}}+infty.
        $$

        If $b>1$ then $b^x to+infty$ as $xtoinfty$, so we may apply L'Hôpital's rule.
        Observing that $b^x = expbig(xlog (b)big)$, we'll obtain that
        $$
        lim_{xtoinfty}frac{x^a}{b^x} =
        lim_{xtoinfty} frac{a}{log(b)}frac{x^{a-1}}{b^x}
        $$

        Notice that up to the positive constant $a/log (b)$, we arrive at the same limit we began with, except the initial value of the exponent $a$ in the polynomial term has decreased by $1$.
        We can successively apply L'Hôpital's rule until the exponent becomes non-positive, at which point the limit is easily seen to be $0$. $square$





        For your particular question, we can apply the general result with $b=frac1a > 1$.






        share|cite|improve this answer



























          up vote
          2
          down vote



          accepted










          Here's a general result.




          Theorem (Domination of exponentials and polynomials):
          Let $a, b>0$.
          Then
          $$
          lim_{xtoinfty}frac{x^a}{b^x} =
          left{
          begin{array}{cc}
          +infty&text{if $,bleqslant 1$}\
          0&text{if $,b> 1$}\
          end{array}
          right.
          $$




          Proof:
          If $bleqslant 1$ then $b^xleqslant 1$ for all $xgeqslant 1$ and hence for $xgeqslant 1$ we have
          $$
          frac{x^a}{b^x} geqslant x^a stackrel{xtoinfty}{xrightarrow{hspace{1cm}}}+infty.
          $$

          If $b>1$ then $b^x to+infty$ as $xtoinfty$, so we may apply L'Hôpital's rule.
          Observing that $b^x = expbig(xlog (b)big)$, we'll obtain that
          $$
          lim_{xtoinfty}frac{x^a}{b^x} =
          lim_{xtoinfty} frac{a}{log(b)}frac{x^{a-1}}{b^x}
          $$

          Notice that up to the positive constant $a/log (b)$, we arrive at the same limit we began with, except the initial value of the exponent $a$ in the polynomial term has decreased by $1$.
          We can successively apply L'Hôpital's rule until the exponent becomes non-positive, at which point the limit is easily seen to be $0$. $square$





          For your particular question, we can apply the general result with $b=frac1a > 1$.






          share|cite|improve this answer

























            up vote
            2
            down vote



            accepted







            up vote
            2
            down vote



            accepted






            Here's a general result.




            Theorem (Domination of exponentials and polynomials):
            Let $a, b>0$.
            Then
            $$
            lim_{xtoinfty}frac{x^a}{b^x} =
            left{
            begin{array}{cc}
            +infty&text{if $,bleqslant 1$}\
            0&text{if $,b> 1$}\
            end{array}
            right.
            $$




            Proof:
            If $bleqslant 1$ then $b^xleqslant 1$ for all $xgeqslant 1$ and hence for $xgeqslant 1$ we have
            $$
            frac{x^a}{b^x} geqslant x^a stackrel{xtoinfty}{xrightarrow{hspace{1cm}}}+infty.
            $$

            If $b>1$ then $b^x to+infty$ as $xtoinfty$, so we may apply L'Hôpital's rule.
            Observing that $b^x = expbig(xlog (b)big)$, we'll obtain that
            $$
            lim_{xtoinfty}frac{x^a}{b^x} =
            lim_{xtoinfty} frac{a}{log(b)}frac{x^{a-1}}{b^x}
            $$

            Notice that up to the positive constant $a/log (b)$, we arrive at the same limit we began with, except the initial value of the exponent $a$ in the polynomial term has decreased by $1$.
            We can successively apply L'Hôpital's rule until the exponent becomes non-positive, at which point the limit is easily seen to be $0$. $square$





            For your particular question, we can apply the general result with $b=frac1a > 1$.






            share|cite|improve this answer














            Here's a general result.




            Theorem (Domination of exponentials and polynomials):
            Let $a, b>0$.
            Then
            $$
            lim_{xtoinfty}frac{x^a}{b^x} =
            left{
            begin{array}{cc}
            +infty&text{if $,bleqslant 1$}\
            0&text{if $,b> 1$}\
            end{array}
            right.
            $$




            Proof:
            If $bleqslant 1$ then $b^xleqslant 1$ for all $xgeqslant 1$ and hence for $xgeqslant 1$ we have
            $$
            frac{x^a}{b^x} geqslant x^a stackrel{xtoinfty}{xrightarrow{hspace{1cm}}}+infty.
            $$

            If $b>1$ then $b^x to+infty$ as $xtoinfty$, so we may apply L'Hôpital's rule.
            Observing that $b^x = expbig(xlog (b)big)$, we'll obtain that
            $$
            lim_{xtoinfty}frac{x^a}{b^x} =
            lim_{xtoinfty} frac{a}{log(b)}frac{x^{a-1}}{b^x}
            $$

            Notice that up to the positive constant $a/log (b)$, we arrive at the same limit we began with, except the initial value of the exponent $a$ in the polynomial term has decreased by $1$.
            We can successively apply L'Hôpital's rule until the exponent becomes non-positive, at which point the limit is easily seen to be $0$. $square$





            For your particular question, we can apply the general result with $b=frac1a > 1$.







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Nov 15 at 5:27

























            answered Nov 15 at 5:22









            Fimpellizieri

            17.2k11836




            17.2k11836






















                up vote
                0
                down vote













                Let $x=1+t$ then Bernoulli's inequality shows
                $$lim_{x to infty}{x^aa^x}=lim_{t to infty}{(1+t)^aa^{1+t}}>lim_{t to infty}{(at)a^{1+t}}=a^2lim_{t to infty}{ta^{t}}=0$$
                the last one holds by L'Hospital.






                share|cite|improve this answer

























                  up vote
                  0
                  down vote













                  Let $x=1+t$ then Bernoulli's inequality shows
                  $$lim_{x to infty}{x^aa^x}=lim_{t to infty}{(1+t)^aa^{1+t}}>lim_{t to infty}{(at)a^{1+t}}=a^2lim_{t to infty}{ta^{t}}=0$$
                  the last one holds by L'Hospital.






                  share|cite|improve this answer























                    up vote
                    0
                    down vote










                    up vote
                    0
                    down vote









                    Let $x=1+t$ then Bernoulli's inequality shows
                    $$lim_{x to infty}{x^aa^x}=lim_{t to infty}{(1+t)^aa^{1+t}}>lim_{t to infty}{(at)a^{1+t}}=a^2lim_{t to infty}{ta^{t}}=0$$
                    the last one holds by L'Hospital.






                    share|cite|improve this answer












                    Let $x=1+t$ then Bernoulli's inequality shows
                    $$lim_{x to infty}{x^aa^x}=lim_{t to infty}{(1+t)^aa^{1+t}}>lim_{t to infty}{(at)a^{1+t}}=a^2lim_{t to infty}{ta^{t}}=0$$
                    the last one holds by L'Hospital.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Nov 15 at 5:15









                    Nosrati

                    26.2k62352




                    26.2k62352






















                        up vote
                        0
                        down vote













                        Consider
                        $$y=a^x ,x^a implies log(y)=a log(x)+x log(a)=x left(afrac{ log (x)}{x}+log (a)right)$$ Since $xto infty$, $frac{ log (x)}{x}$ is negligible for any $a$. So
                        $$log(y) sim x log(a)$$ and, since $0 < a < 1$, $log(a) <0$. Then $log(y)to -infty$ and $y=e^{log(y)} to 0$






                        share|cite|improve this answer

























                          up vote
                          0
                          down vote













                          Consider
                          $$y=a^x ,x^a implies log(y)=a log(x)+x log(a)=x left(afrac{ log (x)}{x}+log (a)right)$$ Since $xto infty$, $frac{ log (x)}{x}$ is negligible for any $a$. So
                          $$log(y) sim x log(a)$$ and, since $0 < a < 1$, $log(a) <0$. Then $log(y)to -infty$ and $y=e^{log(y)} to 0$






                          share|cite|improve this answer























                            up vote
                            0
                            down vote










                            up vote
                            0
                            down vote









                            Consider
                            $$y=a^x ,x^a implies log(y)=a log(x)+x log(a)=x left(afrac{ log (x)}{x}+log (a)right)$$ Since $xto infty$, $frac{ log (x)}{x}$ is negligible for any $a$. So
                            $$log(y) sim x log(a)$$ and, since $0 < a < 1$, $log(a) <0$. Then $log(y)to -infty$ and $y=e^{log(y)} to 0$






                            share|cite|improve this answer












                            Consider
                            $$y=a^x ,x^a implies log(y)=a log(x)+x log(a)=x left(afrac{ log (x)}{x}+log (a)right)$$ Since $xto infty$, $frac{ log (x)}{x}$ is negligible for any $a$. So
                            $$log(y) sim x log(a)$$ and, since $0 < a < 1$, $log(a) <0$. Then $log(y)to -infty$ and $y=e^{log(y)} to 0$







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered Nov 15 at 5:29









                            Claude Leibovici

                            116k1156131




                            116k1156131






























                                 

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