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Homework Help: Evaluate the following limits if it exist

  1. Dec 12, 2013 #1
    1. The problem statement, all variables and given/known data

    evaluate the following limits if it exist
    lim x sin1/x and limit x sin 1/x
    x→0 x→∞
    2. Relevant equations

    3. The attempt at a solution
    Someone have told me that I should let t=1/x and rewrite the limits.However, once I rewrite the limit, I still cannot evaluate the limits.
    So,how can I find the limits?
  2. jcsd
  3. Dec 12, 2013 #2


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    Gold Member

    First rewrite the expression as $$\lim_{x \rightarrow 0} \frac{\sin\left(1/x\right)}{1/x}$$ and then the reason for introducing the variable ##t## becomes more obvious.
  4. Dec 12, 2013 #3


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    Science Advisor

    You should have learned the limit [tex]\lim_{\theta\to 0}\frac{sin(\theta)}{\theta}[/tex] in first semester Calculus. If you do not remember it, look in a Calculus text in the section on the derivative of sin(x).
  5. Dec 12, 2013 #4


    Staff: Mentor

    The OP's problem statement wasn't very clear, but there are two problems here.

    $$\lim_{x \to 0} x sin(1/x)$$
    $$\lim_{x \to \infty} x sin(1/x)$$

    The hint applies to the second problem.
  6. Dec 12, 2013 #5
    I have used this method,but I still cannot find correct answer that I got the answer is 1.
  7. Dec 12, 2013 #6
    Why limx→0xsin(1/x)is the same as limx→0sin(1/x)1/x ?:confused::shy:
  8. Dec 12, 2013 #7


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    Staff Emeritus
    Science Advisor
    Gold Member

    haha, the only thing they did to rewrite the limit like that is
    [tex] x = \frac{1}{1/x} [/tex]
  9. Dec 12, 2013 #8


    Staff: Mentor

    It isn't the same. What you wrote is sin(1/x) * 1/x. What it should be is sin(1/x)/(1/x).
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