# Thread: Sketching a graph from limt information

1. ## 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.

$\displaystyle \lim \limits_{x \to -\infty} f(x)=2$
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.

$\displaystyle \lim \limits_{x \to -1^-} f(x)=\infty$
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?

$\displaystyle \lim \limits_{x \to -1^+} f(x)=5$
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?

$\displaystyle \lim \limits_{x \to 2^-} f(x)=-\infty$
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.

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

$\displaystyle \lim \limits_{x \to \infty} f(x)=-2$
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.

2. ## Re: Sketching a graph from limt information

Here is one variant.

3. ## Re: Sketching a graph from limt information

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

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

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

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

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

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

I believe the graph loioks like this . . .

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

4. ## 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.