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    Hello to everyone
    I have tried to make a short exercise, but it appears difficult in the end.
    I put the exercise and what i have made here.
    If someone can help me, i will be so thankful.
    Best regard to everyone

    Exercise: Tell if the function x →x^(1+1/x) is defined in (R*)+, study its variations, its limits
    to the terminals and to say if it admits a continuous extension on R +. Finally, say if it admits a
    Extension on R +.


    My answer :

    - We know that 1/x is defined on R*

    -We know that x is defined on R so x is also defined on R+.

    Then x^(1+1/x) is defined on (R*)+



    -We know 1/x is decreasing on ]0;+∞[

    So 1+ 1/x is decreasing



    -We know x is increasing on ]0 ; +∞[



    Then the global function x x^(1+1/x) is decreasing on (R*)+




    -lim 1+1/x= 1

    x→+∞




    -lim x=1

    x→1




    Then lim x^(1+1/x) =1

    x→+∞


    -lim 1+ 1/x= +∞

    x→0




    -lim x=+∞

    x→+∞



    Then lim x^(1+1/x) =+∞

    x→+∞




    let's see if f admits a continuous extension on R+

    f(x) is extensible by continuity in 0 if f(x) has a right or left limit next to 0.
    We extend f by continuity by creating a function f' equal to f on R + such that f'(0)=0

    Yeah that's the problem I don't know how to continue
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    (Original post by PotterGranger)
    Hello to everyone
    I have tried to make a short exercise, but it appears difficult in the end.
    I put the exercise and what i have made here.
    If someone can help me, i will be so thankful.
    Best regard to everyone

    Exercise: Tell if the function x →x^(1+1/x) is defined in (R*)+, study its variations, its limits
    to the terminals and to say if it admits a continuous extension on R +. Finally, say if it admits a
    Extension on R +.


    My answer :

    - We know that 1/x is defined on R*

    -We know that x is defined on R so x is also defined on R+.

    Then x^(1+1/x) is defined on (R*)+



    -We know 1/x is decreasing on ]0;+∞[

    So 1+ 1/x is decreasing



    -We know x is increasing on ]0 ; +∞[



    Then the global function x x^(1+1/x) is decreasing on (R*)+




    -lim 1+1/x= 1

    x→+∞




    -lim x=1

    x→1




    Then lim x^(1+1/x) =1

    x→+∞


    -lim 1+ 1/x= +∞

    x→0




    -lim x=+∞

    x→+∞



    Then lim x^(1+1/x) =+∞

    x→+∞




    let's see if f admits a continuous extension on R+

    f(x) is extensible by continuity in 0 if f(x) has a right or left limit next to 0.
    We extend f by continuity by creating a function f' equal to f on R + such that f'(0)=0

    Yeah that's the problem I don't know how to continue
    Since the limit as x approaches 0 of f(x) is not defined, there is no continuous extension.
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    (Original post by HapaxOromenon3)
    Since the limit as x approaches 0 of f(x) is not defined, there is no continuous extension.
    Isn't the limit at 0 from the right equal to 0?
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    (Original post by RichE)
    Isn't the limit at 0 from the right equal to 0?
    If the limit exists from both sides then you define the continuous extension piecewise as equal to the original function for all values of x other than the discontinuity, and equal to the numerical value of the limit when x is the discontinuous point.
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    (Original post by HapaxOromenon3)
    If the limit exists from both sides then you define the continuous extension piecewise as equal to the original function for all values of x other than the discontinuity, and equal to the numerical value of the limit when x is the discontinuous point.
    Not sure what your point is. In the above example we have a function defined for positive x.

    It has a limit of 0 at 0 so can be continuously extended to nonnegative x.

    It can be extended continuously to the real line by making it an even function.

    PS that said I may have misunderstood the OP's set notation which is pretty nonstandard.
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    (Original post by RichE)
    Not sure what your point is. In the above example we have a function defined for positive x.

    It has a limit of 0 at 0 so can be continuously extended to nonnegative x.

    It can be extended continuously to the real line by making it an even function.
    Or, indeed just by making it 0 for x<=0.

    PS that said I may have misunderstood the OP's set notation which is pretty nonstandard.
    Glad it's not just me!

    I hate to say it, but I'm also a little skeptical that this is a question the OP "tried to make up as a short exercise", which makes me reticent to give too much help.
 
 
 
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