Prove or find a counterexample [duplicate]

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  • If a function has a finite limit at infinity, does that imply its derivative goes to zero?

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Suppose $f(x)$ is bounded and differentiable on $[0,infty)$. Consider the statement:



If $lim_xto inftyf(x)=0 $, then $lim_xto inftyf'(x)=0 $.



Prove it if it's right or show a counterexample.




I think it's not true and trying to find a counterexample by using functions containing $cos(frac1x)$ or $e^-x$ or other things. The final goal is to find something make it's derivative' limit at infite doesn't exist. Since if it exists, it must equals zero or $f(x)$ will not be bounded.




calculus examples-counterexamples




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marked as duplicate by Nosrati, leonbloy, Paul Frost, Ethan Bolker, Community♦ Aug 31 at 6:53


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  • Related (dup, I'd say) math.stackexchange.com/questions/162078
    – leonbloy
    Aug 30 at 20:03















up vote
5
down vote

favorite













This question already has an answer here:



  • If a function has a finite limit at infinity, does that imply its derivative goes to zero?

    6 answers




Suppose $f(x)$ is bounded and differentiable on $[0,infty)$. Consider the statement:



If $lim_xto inftyf(x)=0 $, then $lim_xto inftyf'(x)=0 $.



Prove it if it's right or show a counterexample.




I think it's not true and trying to find a counterexample by using functions containing $cos(frac1x)$ or $e^-x$ or other things. The final goal is to find something make it's derivative' limit at infite doesn't exist. Since if it exists, it must equals zero or $f(x)$ will not be bounded.




calculus examples-counterexamples




share|cite|improve this question














marked as duplicate by Nosrati, leonbloy, Paul Frost, Ethan Bolker, Community♦ Aug 31 at 6:53


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














  • Related (dup, I'd say) math.stackexchange.com/questions/162078
    – leonbloy
    Aug 30 at 20:03













up vote
5
down vote

favorite









up vote
5
down vote

favorite












This question already has an answer here:



  • If a function has a finite limit at infinity, does that imply its derivative goes to zero?

    6 answers




Suppose $f(x)$ is bounded and differentiable on $[0,infty)$. Consider the statement:



If $lim_xto inftyf(x)=0 $, then $lim_xto inftyf'(x)=0 $.



Prove it if it's right or show a counterexample.




I think it's not true and trying to find a counterexample by using functions containing $cos(frac1x)$ or $e^-x$ or other things. The final goal is to find something make it's derivative' limit at infite doesn't exist. Since if it exists, it must equals zero or $f(x)$ will not be bounded.




calculus examples-counterexamples




share|cite|improve this question















This question already has an answer here:



  • If a function has a finite limit at infinity, does that imply its derivative goes to zero?

    6 answers




Suppose $f(x)$ is bounded and differentiable on $[0,infty)$. Consider the statement:



If $lim_xto inftyf(x)=0 $, then $lim_xto inftyf'(x)=0 $.



Prove it if it's right or show a counterexample.




I think it's not true and trying to find a counterexample by using functions containing $cos(frac1x)$ or $e^-x$ or other things. The final goal is to find something make it's derivative' limit at infite doesn't exist. Since if it exists, it must equals zero or $f(x)$ will not be bounded.





This question already has an answer here:



  • If a function has a finite limit at infinity, does that imply its derivative goes to zero?

    6 answers





calculus examples-counterexamples





share|cite|improve this question













share|cite|improve this question




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edited Aug 30 at 17:15









Clayton

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18.3k22883










asked Aug 30 at 17:08









Jaqen Chou

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3499




marked as duplicate by Nosrati, leonbloy, Paul Frost, Ethan Bolker, Community♦ Aug 31 at 6:53


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.






marked as duplicate by Nosrati, leonbloy, Paul Frost, Ethan Bolker, Community♦ Aug 31 at 6:53


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.













  • Related (dup, I'd say) math.stackexchange.com/questions/162078
    – leonbloy
    Aug 30 at 20:03

















  • Related (dup, I'd say) math.stackexchange.com/questions/162078
    – leonbloy
    Aug 30 at 20:03
















Related (dup, I'd say) math.stackexchange.com/questions/162078
– leonbloy
Aug 30 at 20:03





Related (dup, I'd say) math.stackexchange.com/questions/162078
– leonbloy
Aug 30 at 20:03











1 Answer
1






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up vote
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accepted










You can take $f(x)=e^-xsin(e^x)$, for instance. Note that $f'(x)=-e^-xsin(e^x)+cos(e^x)$ and that therefore the limite $lim_xto+inftyf'(x)$ does not exist.






share|cite|improve this answer





























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    14
    down vote



    accepted










    You can take $f(x)=e^-xsin(e^x)$, for instance. Note that $f'(x)=-e^-xsin(e^x)+cos(e^x)$ and that therefore the limite $lim_xto+inftyf'(x)$ does not exist.






    share|cite|improve this answer


























      up vote
      14
      down vote



      accepted










      You can take $f(x)=e^-xsin(e^x)$, for instance. Note that $f'(x)=-e^-xsin(e^x)+cos(e^x)$ and that therefore the limite $lim_xto+inftyf'(x)$ does not exist.






      share|cite|improve this answer
























        up vote
        14
        down vote



        accepted







        up vote
        14
        down vote



        accepted






        You can take $f(x)=e^-xsin(e^x)$, for instance. Note that $f'(x)=-e^-xsin(e^x)+cos(e^x)$ and that therefore the limite $lim_xto+inftyf'(x)$ does not exist.






        share|cite|improve this answer














        You can take $f(x)=e^-xsin(e^x)$, for instance. Note that $f'(x)=-e^-xsin(e^x)+cos(e^x)$ and that therefore the limite $lim_xto+inftyf'(x)$ does not exist.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Aug 30 at 18:40









        Kamil Maciorowski

        2,2711819




        2,2711819










        answered Aug 30 at 17:11









        José Carlos Santos

        120k16101182




        120k16101182












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