How do I compute this limit?

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For the following function $f$ and point $a$, determine whether $lim limits_x to a f (x)$ exists, and compute the limit if it exists. Justify your answer.



$f(x)=cos(frac1(1-x)^2)$ where $a=1$



This is a 3 mark question on a past paper so it shouldn't be too complex. I suspect you have to manipulate the $frac1(1-x)^2$ somehow, but I just can't see it.




calculus real-analysis limits




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  • 1




    What happens to the inside as $x to 1$?
    – Randall
    Aug 6 at 15:56






  • 1




    @Dzoooks Not really, since the inside blows up.
    – Randall
    Aug 6 at 15:58






  • 2




    Use geogebra or desmos to plot out the function graph, put in values of x closer and closer to the limit from either side. What are your findings?
    – Nick
    Aug 6 at 16:10














up vote
0
down vote

favorite












For the following function $f$ and point $a$, determine whether $lim limits_x to a f (x)$ exists, and compute the limit if it exists. Justify your answer.



$f(x)=cos(frac1(1-x)^2)$ where $a=1$



This is a 3 mark question on a past paper so it shouldn't be too complex. I suspect you have to manipulate the $frac1(1-x)^2$ somehow, but I just can't see it.




calculus real-analysis limits




share|cite|improve this question

















  • 1




    What happens to the inside as $x to 1$?
    – Randall
    Aug 6 at 15:56






  • 1




    @Dzoooks Not really, since the inside blows up.
    – Randall
    Aug 6 at 15:58






  • 2




    Use geogebra or desmos to plot out the function graph, put in values of x closer and closer to the limit from either side. What are your findings?
    – Nick
    Aug 6 at 16:10












up vote
0
down vote

favorite









up vote
0
down vote

favorite











For the following function $f$ and point $a$, determine whether $lim limits_x to a f (x)$ exists, and compute the limit if it exists. Justify your answer.



$f(x)=cos(frac1(1-x)^2)$ where $a=1$



This is a 3 mark question on a past paper so it shouldn't be too complex. I suspect you have to manipulate the $frac1(1-x)^2$ somehow, but I just can't see it.




calculus real-analysis limits




share|cite|improve this question













For the following function $f$ and point $a$, determine whether $lim limits_x to a f (x)$ exists, and compute the limit if it exists. Justify your answer.



$f(x)=cos(frac1(1-x)^2)$ where $a=1$



This is a 3 mark question on a past paper so it shouldn't be too complex. I suspect you have to manipulate the $frac1(1-x)^2$ somehow, but I just can't see it.





calculus real-analysis limits





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share|cite|improve this question




share|cite|improve this question








edited Aug 6 at 16:04









gimusi

65.4k73684




65.4k73684









asked Aug 6 at 15:54









user499701

777




777







  • 1




    What happens to the inside as $x to 1$?
    – Randall
    Aug 6 at 15:56






  • 1




    @Dzoooks Not really, since the inside blows up.
    – Randall
    Aug 6 at 15:58






  • 2




    Use geogebra or desmos to plot out the function graph, put in values of x closer and closer to the limit from either side. What are your findings?
    – Nick
    Aug 6 at 16:10












  • 1




    What happens to the inside as $x to 1$?
    – Randall
    Aug 6 at 15:56






  • 1




    @Dzoooks Not really, since the inside blows up.
    – Randall
    Aug 6 at 15:58






  • 2




    Use geogebra or desmos to plot out the function graph, put in values of x closer and closer to the limit from either side. What are your findings?
    – Nick
    Aug 6 at 16:10







1




1




What happens to the inside as $x to 1$?
– Randall
Aug 6 at 15:56




What happens to the inside as $x to 1$?
– Randall
Aug 6 at 15:56




1




1




@Dzoooks Not really, since the inside blows up.
– Randall
Aug 6 at 15:58




@Dzoooks Not really, since the inside blows up.
– Randall
Aug 6 at 15:58




2




2




Use geogebra or desmos to plot out the function graph, put in values of x closer and closer to the limit from either side. What are your findings?
– Nick
Aug 6 at 16:10




Use geogebra or desmos to plot out the function graph, put in values of x closer and closer to the limit from either side. What are your findings?
– Nick
Aug 6 at 16:10










2 Answers
2






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Let consider as $nto infty$



  • $x_n=1-sqrtfrac1npito 1 implies f(x_n)=cos(npi)=begincases 1quad textfor n even\-1quad textfor n oddendcases$

then there exist two subsequences with different limit and therefore the limit doesn't exist.






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    Hint:$$beginalignedlim_xto 1cosleft(frac1(1-x)^2right)&=lim_xto 0cosleft(frac1x^2right)\
    &=lim_xto pminftycosleft(x^2right)endaligned$$






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      2 Answers
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      2 Answers
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      up vote
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      down vote













      Let consider as $nto infty$



      • $x_n=1-sqrtfrac1npito 1 implies f(x_n)=cos(npi)=begincases 1quad textfor n even\-1quad textfor n oddendcases$

      then there exist two subsequences with different limit and therefore the limit doesn't exist.






      share|cite|improve this answer

























        up vote
        3
        down vote













        Let consider as $nto infty$



        • $x_n=1-sqrtfrac1npito 1 implies f(x_n)=cos(npi)=begincases 1quad textfor n even\-1quad textfor n oddendcases$

        then there exist two subsequences with different limit and therefore the limit doesn't exist.






        share|cite|improve this answer























          up vote
          3
          down vote










          up vote
          3
          down vote









          Let consider as $nto infty$



          • $x_n=1-sqrtfrac1npito 1 implies f(x_n)=cos(npi)=begincases 1quad textfor n even\-1quad textfor n oddendcases$

          then there exist two subsequences with different limit and therefore the limit doesn't exist.






          share|cite|improve this answer













          Let consider as $nto infty$



          • $x_n=1-sqrtfrac1npito 1 implies f(x_n)=cos(npi)=begincases 1quad textfor n even\-1quad textfor n oddendcases$

          then there exist two subsequences with different limit and therefore the limit doesn't exist.







          share|cite|improve this answer













          share|cite|improve this answer



          share|cite|improve this answer











          answered Aug 6 at 15:58









          gimusi

          65.4k73684




          65.4k73684




















              up vote
              3
              down vote













              Hint:$$beginalignedlim_xto 1cosleft(frac1(1-x)^2right)&=lim_xto 0cosleft(frac1x^2right)\
              &=lim_xto pminftycosleft(x^2right)endaligned$$






              share|cite|improve this answer

























                up vote
                3
                down vote













                Hint:$$beginalignedlim_xto 1cosleft(frac1(1-x)^2right)&=lim_xto 0cosleft(frac1x^2right)\
                &=lim_xto pminftycosleft(x^2right)endaligned$$






                share|cite|improve this answer























                  up vote
                  3
                  down vote










                  up vote
                  3
                  down vote









                  Hint:$$beginalignedlim_xto 1cosleft(frac1(1-x)^2right)&=lim_xto 0cosleft(frac1x^2right)\
                  &=lim_xto pminftycosleft(x^2right)endaligned$$






                  share|cite|improve this answer













                  Hint:$$beginalignedlim_xto 1cosleft(frac1(1-x)^2right)&=lim_xto 0cosleft(frac1x^2right)\
                  &=lim_xto pminftycosleft(x^2right)endaligned$$







                  share|cite|improve this answer













                  share|cite|improve this answer



                  share|cite|improve this answer











                  answered Aug 6 at 16:04









                  Jam

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