Limits of fractions of polynomials and trig functions

  1. I have two...

    1. The problem statement, all variables and given/known data
    The the limit


    2. Relevant equations
    [itex]\lim_{x \rightarrow 1} \frac{1-cosx}{x^2}[/itex]


    3. The attempt at a solution
    I figured to just plug in 1, but I wanted to make sure....

    1. The problem statement, all variables and given/known data
    Find the limit


    2. Relevant equations
    [itex]\lim_{x \rightarrow 3} \frac{\sqrt{x^2-6x+9}}{x-3}[/itex]


    3. The attempt at a solution
    I plugged in the 3, and got 3/0, then I got lost...
     
  2. jcsd
  3. EnumaElish

    EnumaElish 2,483
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    In 2, did you try simplifying the numerator? (What are the roots of the polynomial?)
     
  4. Yes, I tried doing that.

    [itex](x-3)(x-3)[/itex]

    However, I forgot how to get rid of that radical. Squaring wouldn't work, so I have no idea.


    Also, no thoughts on the first one?
     
  5. EnumaElish

    EnumaElish 2,483
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    What is a short hand expression for (x-3)(x-3)?
     
  6. (x-3}^2. Oh yeah, so that takes away the square root, and after everything, it leaves 0. thank you.
     
  7. Avodyne

    Avodyne 1,321
    Science Advisor

    In the first one, are you sure the problem isn't x->0 instead of x->1 ?
     
  8. it is 1, not 0.
     
  9. arildno

    arildno 12,015
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    Gold Member

    Well, then your book has a typo..
     
  10. it's not from my book, it was my teacher.
     
  11. Avodyne

    Avodyne 1,321
    Science Advisor

    Well, it's 99% certain that your teacher meant to write 0 instead of 1. With 1, it's trivial, since both the numerator and denominator are finite, nonzero constants in that limit.
     
  12. arildno

    arildno 12,015
    Science Advisor
    Homework Helper
    Gold Member

    Then he either blundered, or tried to fool you.

    Your function is defined&continuous on all values of x except x=0.

    Your original approach is perfetly valid in the case of x=1.
     
  13. allright, thankyou.
     
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