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Is this graph defined at x = 0?

  1. Dec 14, 2015 #1
    Hello everyone,
    Is the graph ## (2^x - 1) / x ## defined at x = 0?
    When I plot this graph there is no spike at 0 because 0/0 is undefined? Is the computer unable to show this? I am finding the limit as x approaches 0. From the graph limit exits but does f(0) exist?

    Sehr Danke.
     
  2. jcsd
  3. Dec 14, 2015 #2

    Drakkith

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    Staff: Mentor

    The function f(x)=1/x is also undefined at x=0, but there's a spike in the graph of that function because the limit as x approaches 0 is infinity (or negative infinity if coming from the negative side of zero). There's no spike in (2x-1)/x because the limit as x approaches zero is not infinity.
     
  4. Dec 14, 2015 #3

    HallsofIvy

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    Staff Emeritus
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    I would say that it is not a question of the "graph" being defined but the function itself. [itex]f(x)= (2^x- 1)/x[/itex] is "not defined" because "0/0" does not correspond to a number. There is what is usually called a "removable" discontinuity at x= 0. "Removable" because the limit, as x goes to 0, of f(x) exists- it is ln(2). The function "[itex]g(x)= (2^x- 1)/x[/itex] if x is not 0, g(0)= ln(x)" is continuous for all x and is exactly the same as f for all x except 0.
     
    Last edited by a moderator: Dec 14, 2015
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