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Homework Help: A couple of limit computations

  1. Dec 9, 2009 #1
    1. The problem statement, all variables and given/known data
    Compute the limits. If they don't exist, then explain.

    1) lim as x approaches 0 [cos(x) - 1] / [sin^2(x) + x^3]

    2) lim as x approaches 0 √(x^2 + x^3) / (x + 2x^2)


    2. Relevant equations



    3. The attempt at a solution
    1) I replaced the sin^2x in the denominator with 1 - cos^2x and tried to work from there, but I couldn't get anything to cancel.

    2) I couldn't figure out anything on this one either. I factored out an x in the denominator, but again, that didn't lead anywhere.

    We are doing a semester review for the final, and I am starting to get a little nervous, because I don't remember struggling with limit computations when we learned them. Any help that can get me going the right direction would be greatly appreciated. Thank you.

    Edit: This course was taught integrals first, then derivatives. We have not learned L'hospitals rule yet.
     
    Last edited: Dec 9, 2009
  2. jcsd
  3. Dec 9, 2009 #2
    for the second one, did you factor out an x in the numerator?
    If not, that would greatly help
    knowing l'hopital's rule would also really help for the first one
     
  4. Dec 9, 2009 #3
    i know you can't use l'hopital's rule, but i get -1/2 for the first one using l'hopital's rule
     
  5. Dec 9, 2009 #4

    ideasrule

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    Homework Helper

    Have you learned the Taylor series? I'm guessing "no", but your course seems to be ordered in a weird way.
     
  6. Dec 9, 2009 #5
    That's the same thing i thought. I first learned about derivatives(Calc I) then about integrals(Calc II)
     
  7. Dec 9, 2009 #6
    Thanks, I figured out the first one. I factored out an x^2 in the numerator. Giving me
    √[x^2 (1+x)] / x(1+2x)
    ==> x √(1+x) / x(1+2x)
    ==> lim x --> 0 is equal to 1
     
  8. Dec 9, 2009 #7
    that's what I got for the second one.
    so for the first one, i have no clue how to do it without using l'hopital's rule
    did you make any progress on that one?
     
  9. Dec 9, 2009 #8
    That's the way I learned them in high school: derivatives -> integrals. But we are using apostol's calculus, and he goes in the order they were discovered historically.
     
  10. Dec 9, 2009 #9
    No progress on the first one at all. I might just try to use L'hopital's rule anyways. We stopped for the semester one chapter before the book goes over it.
     
  11. Dec 9, 2009 #10
    cool, never heard of it, but the only problem I see in the first one is that x^3.
    If that wasn't there, I probably could have applied the limit in about two to three steps
     
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