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Math Help - Sketching a graph from limt information

  1. #1
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    Sketching a graph from limit information

    I've been struggling with this problem for, I kid you not, a week off and on. I get part of it sketched but then I think I've contradicted myself and I start second guessing. I suppose I'll write what I think it says and correct me based on that.

    \lim \limits_{x \to -\infty} f(x)=2
    I take this to mean that as x approaches y=2, we drop down to negative infinity. I've drawn a horizontal asymptote here but I'm unsure if that's correct.


    \lim \limits_{x \to -1^-} f(x)=\infty
    As we get to negative 1 from the left, go north. This means we never actually hit -1 from the right, we just approach it infinitly right?

    \lim \limits_{x \to -1^+} f(x)=5
    Does this mean from the other side we start drawing a line at point (-1,5) and go where? This is where after we go right a bit... This must join up to something else.

    f is continuous from the right at -1
    Does this information pairs up with the limit just above this one?

    \lim \limits_{x \to 2^-} f(x)=-\infty
    At this point I'm lost. I think I've messed something up before here and by the time I get to this point it's game over.

    \lim \limits_{x \to 2^+} f(x)=+\infty
    Second last piece of information.

    \lim \limits_{x \to \infty} f(x)=-2
    Just draw an asymptote here right?

    If time permits please show me how to appraoch this type of question. What should I look for first? What should I tackle last? I double checked all the equations, hopefully there are no typos.
    Last edited by Maskawisewin; July 16th 2012 at 08:56 AM.
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  2. #2
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    Re: Sketching a graph from limt information

    Here is one variant.

    Sketching a graph from limt information-graph1.png
    Thanks from Maskawisewin
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  3. #3
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    Re: Sketching a graph from limt information

    Hello, Maskawisewin!

    \lim_{x\to \text{-}\infty} f(x)\:=\:2
    To the far left, the graph approaches the horizontal line y \,=\,2


    \lim _{x\to\text{-}1^-}f(x)\:=\:\infty
    There is a vertical asymptote at x = -1
    As x approaches -1 from the left, the graph rises to +\infty.



    \lim_{x\to\text{-}1^+} f(x)\:=\:5
    As x approaches -1 from the right, the graph approaches the point (-1,5).


    \lim_{x\to 2^-} f(x)\:=\:-\infty
    As x approaches 2 from the left, the graph goes down to -\infty.


    \lim_{x\to2^+} f(x)\:=\:+\infty
    As x approaches 2 from the right, the graph goes up to +\infty.


    \lim_{x\to\infty}f(x)\:=\:-2
    As x goes to infinity, the graph approaches the horizontal line y = -2.


    I believe the graph loioks like this . . .

    Code:
                        :     |       :
                       *:     |       :*
                        :     |       :
                      * :     |       :
                  *     o     |       : *
              *         :   * |       :
        *               :     | *     :  *
      - - - - - - - - - + - - +2  *   :   *
                        :     |    *  :     *
    --------------------+-----+-------:--------*-----------------
                        :     |     * :             * 
                        :     |       :                     *
                        :   -2+ - - - : - - - - - - - - - - - -
                        :     |      *:
                        :     |       :

    Thanks from Maskawisewin
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  4. #4
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    Re: Sketching a graph from limt information

    Well no wonder I got confused. I got the left one correct and the and centre one mostly correct but then the third one really screwed me up. I kept thinking the centre and right graphs were existing in the same x axis space (as in not a function, can draw a line through two lines). Thanks for the prompt answers.
    Last edited by Maskawisewin; July 16th 2012 at 09:55 AM.
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