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Limits: Discontinuity

  1. Jun 9, 2008 #1
    1. The problem statement, all variables and given/known data
    Find all values x=a where the function is discontinuous. For each value of x, give the limit of the function as x approaches a.

    2. Relevant equations

    3. The attempt at a solution
    ok the first thing I did was factored out: (x+2)(x-2)/(x-2), then I crossed out the x-2 and ended up with f(x)=x+2...then I checked the answer and the answer was 2,4 I don't understand how they got 2,4 besides that f(x) can't equal 2, because then it'd be undefined.
  2. jcsd
  3. Jun 9, 2008 #2
    When x = 2, function is undefined. (I always confuse continuity concepts, but I think you are not allowed to factor or simplify things)

    and limit of function as it approaches x = 2 is 4

    Two things that question asked for.
  4. Jun 9, 2008 #3


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    Two functions are equal iff they have the same domain, range, and for every x in the domain f(x) = g(x). Thus, if f(x) is your original function, and g(x) = x+2 then f(x) and g(x) are only equal if you restrict g(x) to all real numbers except 2 which is line x+2 with a gap at the point (2,4).
  5. Jun 9, 2008 #4
    I'm still not grasping the concept, I don't understand where we get 2 from besides when 2-2=0, and you can't divide any number into zero.

    Also, I'm trying to work other problems, but apparently, it doesn't make sense, such as

    [5+x]/[x(x-2)], so if I can't change anything as rootX says, how am I supposed to know my next step?
  6. Jun 9, 2008 #5
    for second: discontinous when x(x-2) = 0

    so either x = 0 or x-2 = 0

    for first: without changing anything plug in 2, and you get 0/0 (so discontinouty)
  7. Jun 9, 2008 #6


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    A function f(x) is continuous at x=a if i) f(a) exists and ii) the limit x->a f(x) exists and iii) the limit equals f(a). Your first example is one where the limit exists (here you can cancel things), but f(a) does not exist. The second is one where the limit doesn't exist and neither does f(a). Does that help? And your second one isn't a 0/0 case, that's the first one.
  8. Jun 10, 2008 #7


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    Then go back and read the question again! The question asks for a value of x for which the function is not continuous. It is not continuous at x= 2 precisely because it is undefined there.
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