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Calculating the limit

  1. Apr 17, 2010 #1
    1. The problem statement, all variables and given/known data

    Find the limit of f(x) as x-> - (Infinity)

    2. Relevant equations


    3. The attempt at a solution

    I knew that for rational functions we have to divide both numerator and denominator by the highest power of x in the denominator, but this is confusing. Should I expand the equations? Please help ??
  2. jcsd
  3. Apr 17, 2010 #2
  4. Apr 17, 2010 #3
    I am not allowed to use L'Hopital Rule :cry:
  5. Apr 17, 2010 #4
    That is probably the most direct way to get the answer. Just expand the polynomials out in the numerator and the denominator.

    [tex] lim_{x \to \infty} \frac{8\,{x}^{7}-36\,{x}^{6}+54\,{x}^{5}-27\,{x}^{4}+8\,{x}^{3}-36\,{x}^{2}+54\,x-27}{{x}^{7}+5\,{x}^{6}+8\,{x}^{5}-15\,{x}^{3}-19\,{x}^{2}-10\,x-2}[/tex]

    Then, as x goes to infinity, only the highest order terms matter.

    [tex] =lim_{x \to \infty} \frac{8\,{x}^{7}}{{x}^{7}}[/tex]

    From here, the answer of 8 should be clear.
  6. Apr 17, 2010 #5


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    Your original plan looked ok to me. Imagine multiplying numerator and denominator out, without actually doing it. The highest power will be x^7, right? So divide by x^7. In the numerator split x^7=x^4*x^3. Divide the first factor by the x^4 and the second by x^3. In the denominator split it into x^7=x^5*x^2.
  7. Apr 17, 2010 #6
    Thank you :smile:

    Since this question is based on my calculus exam, I think I won't find time to multiply the whole polynomial. Therefore I will try to split and divide :)
  8. Apr 17, 2010 #7
    That is a good idea. Dick always gives good advice.

    I'd like to point out that, in my suggestion, I was hoping that you would see that once you solve one of these problems, you really don't need to multiply out the whole polynomial. Only the highest order terms matter here. One can get to the step [tex] =lim_{x \to \infty} \frac{-8\,{x}^{7}}{{-x}^{7}} [/tex] by multiplying out the highest order terms in your head.
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