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Limit of function ( sandwich method)

  1. Nov 12, 2008 #1
    limit of function ("sandwich" method)

    1. The problem statement, all variables and given/known data

    Using the "sandwich" method prove that [tex]\lim_{n\rightarrow \propto }(\frac{sin(n)}{n})=0[/tex]


    2. Relevant equations

    [tex]x_n \leq y_n \leq z_n[/tex]

    [tex]\lim_{n\rightarrow \propto }(x_n) \leq \lim_{n\rightarrow \propto }(y_n) \leq \lim_{n\rightarrow \propto }(z_n)[/tex]

    3. The attempt at a solution

    I am honestly little bit confused at this point.

    If the answer is:

    [tex]\frac{-1}{n} \leq \frac{sin(n)}{n} \leq \frac{1}{n}[/tex]

    then my question is if [itex]n=-\frac{\pi}{4}[/itex] then [tex]\frac{-1}{-0.785}[/tex] will be not less or equal to [tex]\frac{\sqrt{2}}{2*(-0.785)}[/tex], where -0.785=[itex]-\frac{\pi}{4}[/itex], where [itex]\pi \approx 3.14[/itex].

    Thanks in advance.
     
    Last edited: Nov 12, 2008
  2. jcsd
  3. Nov 12, 2008 #2

    marcusl

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    Re: limit of function ("sandwich" method)

    Are you sure that n is a real number? Usually n denotes a positive integer in this type of problem.
     
  4. Nov 12, 2008 #3

    HallsofIvy

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    Re: limit of function ("sandwich" method)

    [itex]-1/n\le sin(n)/n\le 1/n[/itex] for n positive. Obviously, if n is negative, just [itex]-1/n\le 1/n[itex] is not true! Your use of [itex]x\rightarrow \propto[/itex] is a little confusing. Did you mean [itex]\infty[/itex]? Even if you do not interpret n as necessarily being positive, if n is "going to [itex]\infty[/itex]" eventually, for some finite N, if n> N, n will be postive. And you can always drop any finite number of terms in an infinite sequence without changing the limit.
     
  5. Nov 13, 2008 #4
    Re: limit of function ("sandwich" method)

    Thanks for the posts. I see now, it was my mistake if an=sin(n)/n, an is progression where n are positive integer numbers. So if:
    [tex]-1 \leq sin(n) \leq 1 [/tex]
    then divided by n, I'll get:
    [tex]-1/n \leq sin(n)/n \leq 1/n[/tex]
    Sorry for the symbol, I misspelled it, since I don't cover LaTeX too good at this moment.
    Thanks for the help.
     
  6. Nov 13, 2008 #5

    HallsofIvy

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    Re: limit of function ("sandwich" method)

    For future reference, in LaTex, [itex]\infty[/itex] is "\infty". [itex]\propto[/itex] is "\propto", i.e. "proportional to".
     
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