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Homework Help: Limits problem

  1. Sep 25, 2008 #1
    1. The problem statement, all variables and given/known data

    lim (x^2-5x+4) / ((sin(x^1/2) - 2))
    x->4



    2. Relevant equations



    3. The attempt at a solution

    Well I factored the top out to be (x-1)(x-4) and if I plug in 4 I get 0 in the numerator. In the denominator, if I plug in 4, I also end up with zero. Im not too sure what Im supposed to do now
     
  2. jcsd
  3. Sep 25, 2008 #2
    [tex]\lim_{x \rightarrow 4}\frac{x^2-5x+4}{sin(\sqrt x-2)}[/tex]

    Are u allowed to apply l'hopitals rule? If so, then it will work nicely!
     
  4. Sep 25, 2008 #3
    No, we cant use l'hopitals rule because we haven't learned it yet.
     
  5. Sep 25, 2008 #4
    Ok, then here it is what just popped into my head,

    i would let [tex] \sqrt x-2=t[/tex] so when x-->4, t-->0

    now

    [tex]\lim_{x \rightarrow 4}\frac{x^2-5x+4}{sin(\sqrt x-2)}=\lim_{x \rightarrow 4}\frac{(x-1)(\sqrt x-2)(\sqrt x+2)}{sin(\sqrt x-2)}=\lim_{t\rightarrow 0}\frac{t}{sin(t)}(t+4)[(t+2)^2-1][/tex]

    now:

    [tex]\lim_{t\rightarrow 0}\frac{t}{sint}=\lim_{t\rightarrow 0}{\frac{1}{\frac{sint}{t}}=1[/tex]

    I think the rest is easy!
     
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