Investigate the function:
and sketch its graph.
For we have a vertical asymptote (for the function becomes ). When we try to find a horizontal asymptote we encounter some problems, because for or the graph approaches . In order to get more information of the behavior of the function for or , we rewrite the function as follows:
We see that the graph approaches:
when or . In fact, this parabola is a special asmptote.
It is clear that the graph has no intersection points with the -axis (). The intersection point with the -axis can be found by solving the equation:
We notice immediately that this equation has the solution and thus we can factorize the function :
The quadratic function has a discriminant which is less than is and thus has no solutions. Thus the only intersection point with the -as is .
Because of the special nature of the function we look at it in more detail near . We notice that the second term near has either the largest (positive or negative) value:
Based on this investigation we can sketch the graph. We also depicted the parabola.