Vertical Asymptotes

 
 

Now assume that the numerator and denominator of f(x) have no common factors,  and discuss what may happen when the values of x are close to the zeros of the denominator.

Behaviour about a vertical asymptote is well illustrated by the example , . Click to see the graph.


Note that f(x) is not defined at x =  0   but is defined for values of x as close as we want to 0.  We know the following:

When x is very close to 0 and  positive,  say  x = .0001,   then f(x) is large   (f(x) =  10000)    and positive.  As x gets even closer to 0, then f(x) becomes even larger while remaining positive.  In fact f(x) can be made as large as we wish in the positive direction by making x sufficiently close to 0 while positive.  This means that when we draw the graph of f(x) = ,  as the values of x approach 0 through positive values, the graph rises and becomes closer and closer to the vertical line x = 0.   In mathematical language,  f(x)  as 

When x is very close to 0 and negative, say x = -.0001, then f(x) is large  in magnitude (f(x) = -10000) and negative.  In fact, f(x) can be made as large as we wish in the negative direction by making x sufficiently close to zero while negative.   As the values of x approach 0 from the left, the graph drops, becoming closer and closer to the vertical line x = 0.  In this case, we write f(x)  as x.
The line x = 0 is an example of a vertical asymptote.

In general, if the magnitude of f(x) becomes infinite (positively or negatively) as x approaches a particular value, a, then the line x = a is a vertical asymptote.


Look again at the graph of f(x) =   to see how this behavior appears in the graph of the function.


The function  has a vertical asymptote x = 2. Click to view the graph.

The function ,  which coincides with the function   for all values of x except 2, has a vertical asymptote x=-2. Click to view the graph.

And the function  has two vertical asymptotes, x = 0 and at  x=-2. Click to view the graph.

We can now summarise the third step in drawing the graph of a rational function:

STEP 3 
Set the denominator of the function equal to 0 and solve for x. 
Evaluate the function close to each solution to determine the sign of the function making sure to check on both sides of the asymptote. 
 
Exercises
In these exercises, you must determine the vertical asymptote(s) of the given function.  If you think , for example, that the function has a vertical asymptote at x = 2, enter 2 in the box.  Determine if f(x)   or f(x)  on the left of the asymptote, and click in the appropriate grey box.    Next determine if f(x)   or f(x) on the  right of the asymptote.