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Math Help - prove a limit

  1. #1
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    prove a limit

    Suppose that ;

    1)  f: R\rightarrow R^2 such that : f(x) =(sinx,cosx)

    2)The Euclidian norm of a vector  v=(u_{1}, u_{2}) is defined as :

     ||v||_{Eu} =\sqrt{ u_{1}^2 +u_{2}^2}

    3) The maxnorm of a vector  v=(u_{1},u_{2}) is defined as :

     ||v||_{max} =  max( |u_{1}|,|u_{2}|)

    Where  u_{1},u_{2} belong to the real Nos R

    Then prove :

     \lim_{x\to 0} f(x) = (0,1) ,with respect to both norms

    I know that i have to prove the following:


    1) given ε>0 ,i got to find a δ>0 ,such that :

    if 0<|x|<δ , then ||(sinx,cosx)-(0,1)||_{Eu}<\epsilon

    2)given ε>0 ,i must find a δ>0 ,such that:

    if 0<|x|<δ ,then ||(sinx,cosx)-(0,1)||_{max}<\epsilon .

    Any suggestions for that delta??
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by alexandros View Post
    Suppose that ;

    1)  f: R\rightarrow R^2 such that : f(x) =(sinx,cosx)

    2)The Euclidian norm of a vector  v=(u_{1}, u_{2}) is defined as :

     ||v||_{Eu} =\sqrt{ u_{1}^2 +u_{2}^2}

    3) The maxnorm of a vector  v=(u_{1},u_{2}) is defined as :

     ||v||_{max} =  max( |u_{1}|,|u_{2}|)

    Where  u_{1},u_{2} belong to the real Nos R

    Then prove :

     \lim_{x\to 0} f(x) = (0,1) ,with respect to both norms

    I know that i have to prove the following:


    1) given ε>0 ,i got to find a δ>0 ,such that :

    if 0<|x|<δ , then ||(sinx,cosx)-(0,1)||_{Eu}<\epsilon

    2)given ε>0 ,i must find a δ>0 ,such that:

    if 0<|x|<δ ,then ||(sinx,cosx)-(0,1)||_{max}<\epsilon .

    Any suggestions for that delta??
    Try looking at when \sin(x),1-\cos(x) intersect. You obviously want to use \arcsin(\varepsilon). But, what must you impose first?
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  3. #3
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    I am sorry i do not understand,but if you suggest a delta i can try to prove the problem

    Then you can explain how you got that delta
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  4. #4
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    Can anyone help me ,please??
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  5. #5
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    perhaps the problem is not derivable??
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