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I By Continuity definition 1/0 is infinity

  1. Mar 16, 2017 #1
    lim 1/x as x->0 is infinity, but the function taking it to infinity is continuous, but for continous functions f(a)= lim f(x) as x->a, so by defininition 1/0 is infinity, what is wrong with this logic?
     
  2. jcsd
  3. Mar 16, 2017 #2

    pwsnafu

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    1/x is not defined at x=0. This means that ##f(a)## does not exist, hence you can't appeal to continuity (continuity requires a function to be defined there).
     
  4. Mar 16, 2017 #3
    a is 0, f=1/x
     
  5. Mar 16, 2017 #4

    pwsnafu

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    Yeah, sorry. I realised and edited my post. You replied just before I finished.
     
  6. Mar 16, 2017 #5
    oh, 1/x, x element of R, and inf not element of R but cardinality, thanks.
     
  7. Mar 16, 2017 #6

    pwsnafu

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    Also, I want to point out
    is not true. The left and right limits are not the same.
     
  8. Mar 16, 2017 #7
    Why not? 1/x coming from the negative side to 0 should yield same result as from the positive?
     
  9. Mar 16, 2017 #8

    PeroK

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    No.The limit from the left is ##-\infty##. For example ##1/(-.01) = -100##
     
  10. Mar 16, 2017 #9

    mfb

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    There is no way to extend the definition of f to x=0 in a continuous way.
     
  11. Mar 16, 2017 #10

    Mark44

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    Take a look at the graph of y = 1/x. There is the worst possible kind of discontinuity at x = 0, with ##\lim_{x \to 0^-} \frac 1 x = -\infty## and ##\lim_{x \to 0^+} \frac 1 x = +\infty##.
     
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