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Lim x to 0 x*sin(1/x)

  1. Jul 27, 2012 #1
    I was wondering why when solving this limit, you are not allowed to do this:

    [itex]{ lim }_{ x\rightarrow 0 }\quad xsin(\frac { 1 }{ x } )[/itex]

    Break this limit into:

    [itex] { lim }_{ x\rightarrow 0 }\quad x\quad *\quad { lim }_{ x\rightarrow 0 }\quad sin(\frac { 1 }{ x } ) [/itex]

    Then, since, sin(1/x) is bounded between -1 and 1, and lim x-> 0 (x) is 0, the answer should be 0.
     
  2. jcsd
  3. Jul 27, 2012 #2
    That's not rigorous enough, because [itex]lim_{x\rightarrow 0}sin\frac{1}{x}[/itex] doesn't exist. But what you can do is say that [itex]-x\leq xsin\frac{1}{x}\leq x[/itex] for all [itex]x\neq 0[/itex], and [itex]lim_{x\rightarrow 0}(-x)=lim_{x\rightarrow 0}(x)=0[/itex], so by the squeeze theorem [itex]lim_{x\rightarrow 0}xsin\frac{1}{x}=0[/itex]
     
  4. Jul 28, 2012 #3
    "That's not rigorous enough, because lim x→0 sin1 x doesn't exist."

    I have to agree with this person. That limit doesn't really make sense. It only loops through a ratio. Because it loops through a ratio along with another variable, it makes the limit of the function an oscillating one, you have to check the derivatives instead.

    1*sin(1/x) - x(-1/x^2)(sin1/x)(cos1/x)
    take apart the trigonometric functions... because they only loop through -1 and 1
    1 + (1/x)
    ... and you get an idea of where the limit is going as approaches 0
    This is showing you that the derivative is unstable. It just does not level off.

    lim(x->0) for sin(1/x) will just continue to oscillate as sin(1/x) flucuates through ratios.


    This should support the person's above statement that the limit doesn't really exist, and that values will always be between 1 and -1.

    I am not so sure however myself. I haven't done some of this kind of work in a year. Can anyone add more input? I can't remember the exact procedure for checking this one's limit.

    The actual limit cannot be zero as a result of the oscillating functions. Just think, if 1/x allowed x to get smaller and smaller, 1/x as a function would continue to grow. It ends up as sin (+t) where t continues to increase without bounds. The sine function still can only go to one. Even when being subjected to proportion adjustments from it's neighbor variable x. Hence, X * sin(+t) cannot have a zero limit, it really looks more like (k)sin(+t) if you think about it. It will never leave the 1*k and -1*k range.

    It is hard to explain, but if you check your calculator it might make a little more sense. Just use .01 sized values for x in the function sin(1/x) Notice that these values when multiplied with the same value of nearby x will still be in flux. Using my own calculator, I see now when checking with .01 sized values that 0 in fact cannot be the limit, check for yourself!

    Punch this is in and say I showed you how...

    xsin(1/x) and then use DELTATBL=.001

    check the tables and you will see exactly what I mean....

    It is from here that you should be able to say that the limit might actually be zero...
     
    Last edited: Jul 28, 2012
  5. Jul 28, 2012 #4

    There's no mathematical sound meaning to [itex]\,\lim f(x)g(x)=\lim f(x)\cdot \lim g(x)\,[/itex] if any of these limits doesn't exist, yet

    you too can tell what you said, namely: if
    [tex]\lim_{x\to\ x_0}f(x)=0\,\,\text{and}\,\,\,g(x)\,\,\,\text{ is bounded in some neighbourhood of}\,\,x_0[/tex]
    then [tex]\lim_{x\to x_0}f(x)g(x)=0[/tex]

    The [itex]\,\epsilon - \delta[/itex] is very simple.

    DonAntonio
     
  6. Jul 28, 2012 #5
    It isn't too hard if you think about it. Consider the sequence {x_n} = {1/pi, 1/2pi, 1/3pi, ...}.
    Then sin(1/x_n) = 0 and thus goes to 0 as n goes to infinity.

    Similarly, consider {y_n} = {2/pi, 2/5pi, 2/9pi, ...}. Then sin(1/y_n) = 1 and thus goes to 0 as n goes to infinity.

    Since you have two subsequences approaching 0 such that the original function approaches two different vales, the limit cannot exist.
     
  7. Jul 28, 2012 #6
    It isn't too hard if you think about it. Consider the sequence {x_n} = {1/pi, 1/2pi, 1/3pi, ...}.
    Then sin(1/x_n) = 0 and thus goes to 0 as n goes to infinity.

    Similarly, consider {y_n} = {2/pi, 2/5pi, 2/9pi, ...}. Then sin(1/y_n) = 1 and thus goes to 0 as n goes to infinity.

    Since you have two subsequences approaching 0 such that the original function approaches two different vales, the limit cannot exist.
     
  8. Jul 28, 2012 #7


    The wanted limit is when [itex]x\to 0[/itex] , not to infinity.

    DonAntonio
     
  9. Jul 28, 2012 #8
    Yes, x_n and y_n are going to 0. The index n is going to infinity. I merely created sequences x_n and y_n that approach 0 such that sin(1/x_n) and sin(1/y_n) approach different values.
     
  10. Jul 28, 2012 #9
    Substitute x = 1/t. Then the limit we want becomes [itex]\displaystyle \lim_{t\to\infty}\frac{\sin(t)}{t}[/itex]. And, sine is a bounded function: [itex]-1 \leq \sin(x) \leq 1[/itex]. This puts the limit in the form of the direct application of the squeeze theorem.

    Can you take it from there?
     
  11. Jul 28, 2012 #10


    It doesn't matter what the sine approaches: it is bounded everywhere and, in particular, it is bounded in any neighbourhoodo f zero. Since the other functions approaches zero the whole thing approaches zero, too.

    DonAnronio
     
  12. Jul 28, 2012 #11
    I know what the OP's original question is. The point of my post was to help illustrate why sin(1/x) does not have a limit as x goes to 0.
     
  13. Jul 28, 2012 #12

    Ok. Perhaps next time it'd a good idea to quote from the OP what you're addressing. In your first post you talked

    about "the original function", which goes to zero, and apparently you were reffering ONLY to part

    of it, namely to [itex]\,\sin 1/x\,[/itex] . This may be confusing.

    DonAntonio.
     
  14. Jul 28, 2012 #13
    I thought it was easily inferred...
     
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