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Possibly the hardest limit question i have ever come across.

  1. Oct 6, 2008 #1
    b]1. The problem statement, all variables and given/known data[/b]
    F(x) = lim csct - cscx find the value of f '(pi/4) ( f prime pi/4)
    t-->x t - x

    (the denominator is screwed up. it should be t-->x for the limit and t-x for the denominator, but you guys probably already guess that. )

    2. Relevant equations
    i dont know. perhaprs first principles. please avoid l'hopitals rule for even i can do that haha. I dont know what im not seeing, i just cant approach the question.


    3. The attempt at a solution
    iv tried hundreds of things. the first thing i tried was multiplying top and bottom by conjugates to see if i could work out something with that. the second thing i tried was changing csc in to its primitive for 1/sinx and finding a lowest common denominator and using fraction rules. the next thing i tried was separating the top and bottom into functions of their own... that just threw me into a brick wall. please if someone can give me a starting point i would appreciate it soooooo much!
     
  2. jcsd
  3. Oct 6, 2008 #2

    Dick

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    Do change csc(x) into 1/sin(x) and rearrange it. I'm guessing you probably know the limit of (sin(t)-sin(x))/(t-x). Use that.
     
  4. Oct 6, 2008 #3
    you are completely right, i could find a solution that way if i was to use l'hopital's rule
    in the ladder part, after all conversions and reductions have been made. however
    i am tending away from his rule because i am in first year calculus and want to
    grasp the "primitive" way of completing the question.
     
  5. Oct 6, 2008 #4

    Dick

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    (sin(t)-sin(x))/(t-x) is a difference quotient for the derivative of the sine function. To do that in a "primitive" way, write it as the limit of (sin(x+h)-sin(x))/h as h->0 and use the trig addition rule. Then use some other "primitive" stuff like lim sin(h)/h=1 as h->0.
     
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