Does lim x →0 sin 1/x exist? also .

  1. Does lim x →0 sin 1/x exist? also.....

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

    does lim x →0 sin(1/x) exist?

    3. The attempt at a solution

    I am very new to calculus. From what i have understood till now, limit of a function at a point exists only if the left and right hand limits are equal. What would the graph of sin(1/x) look like? Would the left and right hand limits exist? Do they exist in all cases, for all functions?
     
  2. jcsd
  3. matt grime

    matt grime 9,395
    Science Advisor
    Homework Helper

    Re: Does lim x →0 sin 1/x exist? also.....

    try drawing it and find out.

    It would be a bit dull if left and right limits always exist. Remember that limits are unique. So, you need to think about sequences of real numbers x_n tending to zero.

    Alternatively we can think about

    lim sin(y)

    as y tends to infinity.
     
  4. Re: Does lim x →0 sin 1/x exist? also.....

    Btw, the existence of left and right limits of a function at a certain point is necessary but not sufficient for a limit to exist at that point. Left and right limits should also be equal for a limit to exist at the same point.
     
  5. HallsofIvy

    HallsofIvy 40,800
    Staff Emeritus
    Science Advisor

    Re: Does lim x →0 sin 1/x exist? also.....

    In particular, think about the sequences [itex]x_n= 1/(n\pi)[/itex], [itex]x_n= 2/((4n+1)\pi)[/itex], and [itex]x_n= 2/((4n-1)\pi)[/itex] a n goes to infinity.
     
  6. Re: Does lim x →0 sin 1/x exist? also.....

    In a real analysis class you would do something like the following:

    Let [tex] f(x) = sin ( \frac {1}{x} ) [/tex]

    Let [tex] s_{n} [/tex] be the sequence [tex] \frac {2}{n \pi} [/tex]

    [tex] \lim_{n \to \infty}s_{n} = 0 [/tex]

    but [tex] (f(s_{n}) )[/tex] is the sequence 1,0,-1,0,1,0,-1,0... which obviously doesn't converge.

    Therefore, [tex] \lim_{x \to 0} sin( \frac {1}{x}) [/tex] does not exist.
     
  7. tiny-tim

    tiny-tim 26,041
    Science Advisor
    Homework Helper

    Hi hermy! :wink:

    matt's :smile: is the best way for general questions like this …

    it shows you what the difficulty is! :wink:
     
  8. Re: Does lim x →0 sin 1/x exist? also.....

    Could you give an example of a function where the left/right hand limit doesn't exist at a point (where the function is defined on the set of all real no.s)?
     
  9. D H

    Staff: Mentor

    Re: Does lim x →0 sin 1/x exist? also.....

    f(x) = 0 if x is rational, 1 if it is irrational.
     
  10. Mark44

    Staff: Mentor

    Re: Does lim x →0 sin 1/x exist? also.....

    Another example is f(x) = 1/x, x [itex]\neq [/itex] 0; f(0) = 0.
    This function is defined for all real numbers. The left hand limit, as x approaches zero is negative infinity, while the right hand limit, as x approaches zero, is positive infinity.
     
  11. Re: Does lim x →0 sin 1/x exist? also.....

    In calculus you need to use the idea of limits. Essentially the limit of sin x/x does equal 1 but you have to show it from both sides.

    If we consider a left hand limit if sin x/x then we can verify the limit with a table of calculations (just grab a calculator and calculate f(x) sin x / x for values approaching x)

    We can also consider the right hand limit also. For the right hand limit we can do the same thing by letting f(x) approach sin x/x.

    Now the limit is only valid if and only if the right hand limit equals the left hand limit. So
    essentially if this is the case and you find a value for it, then you can prove what that limit
    is.
     
  12. D H

    Staff: Mentor

    Re: Does lim x →0 sin 1/x exist? also.....

    The original poster is looking at f(x)=sin(1/x), not g(x)=sin(x)/x.
     
  13. matt grime

    matt grime 9,395
    Science Advisor
    Homework Helper

    Re: Does lim x →0 sin 1/x exist? also.....

    I don't want to give the game away but.... sin(1/x) as x tends to 0 perhaps? I can trivially extend this to a function on the whole of the real line if I want to, if you're worried about it being undefined at zero:

    f(x)=sin(1/x) x=/=0
    f(0)=0 (or anything else you want)
     
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