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Finding a limit without L'Hospital's Rule

  1. Oct 22, 2015 #1
    1. The problem statement, all variables and given/known data
    Hi,
    I have been trying to find the limit of the following expression in order to determine the Big-o(in particular, the order of decaying to zero)

    [tex]\lim_{x\rightarrow 0} \frac {\cos (x) -1 +\frac {x}{2} }{x^4}[/tex]

    Is there another "reasonable" way to solve it?

    Thank you,
    Thomas


    2. Relevant equations
    [tex]\lim_{x\rightarrow 0} \frac {\cos (x) -1 +\frac {x}{2} }{x^4}[/tex]

    3. The attempt at a solution
    I have tried to factorize x^2 in the numerator and denominator, and yield nothing(well, yield that the limit goes to infinity,). tried to replace cosx with sqrt(1-sin^2 x) and multiply with the conjugate but nothing.
     
  2. jcsd
  3. Oct 22, 2015 #2

    Krylov

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    This limit doesn't exist, as can be seen from the Maclaurin series of ##\cos##. Maybe you wanted ##\frac{x^2}{2}## instead of ##\frac{x}{2}##?
     
  4. Oct 22, 2015 #3
    What's a simple equivalent in 0 of ##\cos(x)## ?
     
  5. Oct 22, 2015 #4

    SammyS

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    It would be a much more interesting problem if that was x2/2 in the numerator.

    Are you sure that shouldn't be :
    ##\displaystyle \ \lim_{x\to 0} \frac {\cos (x) -1 +\displaystyle \frac {x^2}{2} }{x^4} \ ##​
     
  6. Oct 22, 2015 #5
    oh! sorry! it is like that!
    it is so embarrassing!

    The Equation is with x^2/2...

    Sorry about that.

    Thomas
     
  7. Oct 22, 2015 #6

    SammyS

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    The quickest way to get a solution is to use the Taylor expansion for cosine.
     
  8. Oct 23, 2015 #7
    Thank you for your answer.

    We have not learned Taylor expansion yet. So, is there a reasonable way to solve it with simple algebraic steps?

    (well, this question is not so important if there is no reasonable way to solve it, so don't feel obliged to give a full solution. Idea will be enough. I have solved it already by L'H., but I am probably a masochist :headbang: )
     
  9. Oct 23, 2015 #8

    SammyS

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    Write the numerator as ##\displaystyle \ \cos (x) +\left(\displaystyle \frac {x^2}{2} -1\right) \ ##.

    Then multiply the numerator & denominator by ##\displaystyle \ \cos (x) -\left(\displaystyle \frac {x^2}{2} -1\right) \,,\ ## to get a difference of squares in the numerator.

    Do some simplifying & see what pops out.
     
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