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Limit of sin x

  1. Oct 24, 2009 #1
    In a sinusoidal function...suppose the value of δ is very large...then as x approaches any a, the value of f(x) might not approach L directly...or there should not be a direct relation; example -

    [tex]\lim_{x \to 1.5} sin x = 0.997494986[/tex]

    Where I've stated δ as 7...then if x = 1.5 – 6.9 = -5.4; as x approach 1.5 from -5.4, value of sin x does not directly approach 0.997494986...it fluctuates between 1 to -1 many times before it reaches that value.

    My question is...is this expression [tex]\lim_{x \to 1.5} sin x = 0.997494986[/tex] with δ as 7 valid?
     
  2. jcsd
  3. Oct 24, 2009 #2

    HallsofIvy

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    It does not matter "how" x approaches a. The only requirement is that "if |x-a|< delta, then |f(x)- L|< epsilon. It is NOT a matter of x getting "closer and closer to a".

    Talking about f(x) changing "as x approaches 1.5", for x distant from 1.5 is completely irrelevant. Given any epsilon> 0, there exist a delta such that if |x- 1.5|< delta, then |sin(x)-0.5381|< epsilon.
     
  4. Oct 24, 2009 #3
    Oh, ok, I get it...I think.

    |sin(x)-0.5381| should not exceed ε if |x- 1.5|< delta.
     
  5. Oct 24, 2009 #4
    Rather the other way around. If |x-1.5| < delta, then |sin(x)-.05381| will be less than epsilon. That's the point of the delta-epsilon proof.
     
  6. Oct 25, 2009 #5
    We can take either ways.
     
  7. Oct 25, 2009 #6

    actually, watch out for the false definition:

    for any epsilon > 0, there exists a delta > 0 such that |f(x) - L | < epsilon => |x-a| < delta

    this is WRONG. it would be a good exercise disproving this
     
  8. Oct 25, 2009 #7
    "B if A" is the same as "if A, then B." If you read carefully, you'll notice dE_logics said the right thing (except with an incorrect value for the limit. I don't know where Halls got 0.5381 from...).
     
  9. Oct 25, 2009 #8

    HallsofIvy

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    Neither do I! I don't know where I got that.
     
  10. Oct 25, 2009 #9
    yeah, I noticed that, but it is good practise to disprove the false statement anyway, many functions work under that particular kind of false definition
     
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