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Limits and continuity test questions

  1. May 2, 2012 #1
    1. The problem statement, all variables and given/known data
    2. Show that the function is continuous on the given interval.
    (a)f(x)= (2x+3)/(x-2) range:(2, infinity)

    (b)f(x) = 1- sqrt(1-x^2) range:[-1,1]

    3. Prove that the following limits do not exist.
    (a) lim x tends to 0 ( absolute|x|/x)

    (b) lim x tends to 3 (2x/(x-3))

    4.Find the limit of the function

    f(x) = x^2 if x=/= 0
    2 if x=0

    at the point (a) x=-1; (b) x=0; (c) x=sqrt(2)

    5. Find all values of a such that f is continuous on (-infinity, infinity):
    (a)
    f(x)= x+1 if x<=a
    x^2 if x>a

    (b) f(x) = (x^2 + a)/(x-1) if x=/=1
    2 if x =1



    3. The attempt at a solution
    Can I answer question 2a and 2b by writing out statements to prove that they are continuous? If that is not allowed, how do I prove that they are continuous?

    I know 2a is continuous on (2, infinity) because 2x + 3 and x - 2 are polynomials and the only point of discontinuity occurs at x=2 as that will cause the denominator to be zero. I also know that for 2b, lim x tends to a f(x) =f(a) when -1<=a<=1 so f(x) is continuous on [-1, 1]. But how do I SHOW it on the paper?

    For question 3a, Can I show that left hand side limit = -1 while right hand side limit = 1, and since left hand side limit is not = to right hand side limit, the limit doesn't exist. For question 3b, do I do the left hand and right hand limit test again or do I just substitute in x=3?

    For question 4, I believe I should just substitute in the x-values when x=-1 and x= sqrt(2). To find the limit for x=0, I could find the left and right hand limit to conclude that lim x tends to 0 is equal to 0. I think that's how it should be done, but it seems too easy to be true.

    As for question 5, I don't have any ideas on how I should start or what I should do, so I hope that someone could guide me on that.

    Many thanks.

    P.S. If my attempt is not informative enough or unsatisfactory, I guess I could try writing it more mathematically.
     
  2. jcsd
  3. May 3, 2012 #2
    You can show 2a and 2b directly with the epsilon-delta-definition of continuity. So calculate what is f(x)-f(y) and show that this can be made arbitrarily small by putting x close to y.

    For 3, your strategy is sound. You can see in 3b that the limit is +infinity from one side and -infinity from another.

    4 is correct.

    For 5, you can probably see that the functions may be discontinuous in a single point. It may help if you just plot the functions and see when they cross at the correct point. So for example in a), you'd plot f(x) = x+1 and g(x) = x^2 and see where they intersect.
     
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