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Stuck on a possibly simple trig limit

  1. Sep 29, 2010 #1
    1. The problem statement, all variables and given/known data

    how do you go about solving the equation
    lim (t->0) 2t / (sin(t)) - t

    the answer in the text is significantly different than what i get.. i can get most of the other trig limits using the fundamental limit etc.. but this one im stuck ? i may be way over complicating it but i think it needs some algebraic manipulation in the current form?

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Sep 29, 2010 #2
    You should notice that [tex]sin \textit{x} /\textit{x} [/tex] is a notable limit
  4. Sep 29, 2010 #3
    lim x->0 of sinx/x =1, so
    lim x->0 of 1/(sinx/x) is also 1, and 1/(sinx/x)=x/sinx, so
    lim x->0 of x/sinx = 1
    from here just rearrange the equation.
  5. Sep 30, 2010 #4
    [STRIKE][STRIKE][/STRIKE][/STRIKE]yes I'm aware of sinx/x = 1 and all the other variations of it etc thats what i was refering to as the fundamental trig limit but i think its the fact its in the form. sinx - x i cannot factor out an x and since its in the form of a binomial this is where i think im just doing something stupid with my algebra any pointers on that one?

    rember its 2t / (sin(t)) - t
  6. Sep 30, 2010 #5
    [tex]2t/ sin(t) - t [/tex]
  7. Sep 30, 2010 #6


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    If you mean

    [tex]\frac{2t}{\sin t -t},[/tex]

    then rewrite it in the form

    [tex]\frac{2}{\frac{\sin t}{t} -1}.[/tex]
  8. Sep 30, 2010 #7
    If you factor out the 2 and split the limit into two parts then you have;
    2 lim t->0 (t/sint) - lim t->0 (t)
  9. Sep 30, 2010 #8
    that was exactly what i was doing wrong.. thank you very much

    it becomes

    forgive my poor latex.. im trying

    [tex]2 / sint/t - t/t[/tex]
    sorry ill have to figure out better latex tomorrow
    but if im correect

    that is 2 / 1 - t/t and since t/t is approaching zero but both equal shouldnt that become 1 as well... giving 1-1 in the denominator and an undefined situation?? the text is saying it should equal DNE or +infinity .... i dont see the infinity situation here?
  10. Sep 30, 2010 #9


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    in the sense that

    [tex]\lim_{x\rightarrow 0} \frac{1}{x} \rightarrow \infty.[/tex]

    So the limit does not exist.
  11. Sep 30, 2010 #10
    but in that form it just purely does not exist am i right? unless you know wether you're approaching zero from the right or the left.. you can end up with -infinity or +infinity....

    so it would need to be a one sided limit.. or am i off?
  12. Sep 30, 2010 #11


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    You're right if the left and right limits don't agree, the limit does not exist. However, it's also true that a divergent result means that the limit doesn't exist.

    In this case, the function is even, so both the right and left limits give [tex]-\infty[/tex]. We still say that the limit does not exist.
  13. Sep 30, 2010 #12
    excellent thanks for your help!... i did look in the text book again to confirm the answer and i see its showing a +infinity?? not sure what the reasoning behind the positive is? but fighting with this question has actually helped me understand a bit deeper. thanks!
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