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Limit questions

  1. Oct 2, 2007 #1
    1. The problem statement, all variables and given/known data

    Find the limit as x approaches x-(x from the left) if f(x) = cscx

    2. Relevant equations

    none

    3. The attempt at a solution

    The only way I can think of solving this is to convert it to 1/sinx which would have a limit of 0. I'm not sure if that even makes any sense though. Any help appreciated.
     
  2. jcsd
  3. Oct 2, 2007 #2

    Dick

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    x approaches what? No matter what it is, the answer isn't zero.
     
  4. Oct 2, 2007 #3
    Sorry the ink was smudged what looked like an x- was actually pi -(pi approaching from the left).
     
  5. Oct 2, 2007 #4

    Dick

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    csc(x)=1/sin(x). If x is approaching pi the denominator is going to zero. The 'limit', such as it is, must be some kind of infinity. Which kind?
     
  6. Oct 2, 2007 #5
    Since it's approaching from the left I would guess negative infinity, but I'm not 100% sure on that.
     
  7. Oct 2, 2007 #6

    Dick

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    Hmm. My guess would be different. But then maybe my left is different from your left.
     
  8. Oct 2, 2007 #7
    I must be confused then because the way I understood it was a limit where "x-->c-" means that we only consider values less than c. So with the limit x-->(pi)- wouldn't it have to be negative infinity since infinity would be a value greater than pi or is my logic completely wrong.
     
  9. Oct 2, 2007 #8

    dynamicsolo

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    Recall that your function is 1/(sin x), so what sign does the denominator have as x approaches pi from lower values?
     
  10. Oct 2, 2007 #9
    Would it be negative since it's approaching a positive value from the left?
     
  11. Oct 2, 2007 #10

    Dick

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    Negative, in the non-affirmative sense. Think about it!
     
  12. Oct 2, 2007 #11
    So would it be a positive value then? That would be the only other option. The reason is why it is positive is what I'm still unsure about.
     
  13. Oct 2, 2007 #12

    Dick

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    What's sin(pi-0.00001). Use a calculator please if you can't draw the graph of sin.
     
  14. Oct 2, 2007 #13
    I get 0.00001 when I enter sin(pi-0.00001) into my calculator.
     
  15. Oct 2, 2007 #14

    Dick

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    Quite reasonable. So what's csc(pi-0.00001) and what happens as x gets even closer to pi?
     
  16. Oct 2, 2007 #15
    I get 100000 and it gets progressively larger and larger the closer it comes to pi, so am I correct to assume the limit is infinity?
     
  17. Oct 3, 2007 #16

    Dick

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    It gets progressively larger and larger. I don't think you have to assume anything. It's infinity. But do you understand why? sin(pi) is zero and to the left of pi, it's positive. So?
     
  18. Oct 3, 2007 #17
    csc is 1/sin, so if sin(pi) is zero a number close to sin(pi) would be a number close zero, so 1/sin(pi) would be 1/(a very small number) and would keep getting larger heading towards infinity. Am I getting close or way off again?
     
  19. Oct 3, 2007 #18

    Dick

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    Yes. Except now say if the number is approaching pi- the very small number is also a very small positive number. So you can call the limit +infinity. If it's pi+ then you want to say -infinity.
     
  20. Oct 3, 2007 #19
    Alright I think I understand now. Thank you for your help.
     
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