When you graph and equation such as  ,
the graph will look like this:
On this graph, you will notice that the curves keep getting closer and
closer to 
and 
.
If you draw vertical lines at these points, and zoom the graph in a little,
you will see this demonstrated.
These two lines,
and
, are called asymptotes. No matter how far you take the graph, it will
never cross these two lines. That is because these two values of x would
make the denominator of the fraction
equal 0, which is undefined.
You can find the vertical asymptotes of an equation by following three
simple steps.
Step 1: Factor the denominator.
Step 2: Set the two factors equal to zero. The zero property of multiplication
says that if any factor in a multiplication problem is zero, the product
will be zero. Therefore, if either 2x + 5 or 7x – 2 equals zero,
the fraction is undefined.
 and 
Step 3: Solve for x
and
Example 1
Here is another example. Find the vertical asymptotes for
Step 1: Factor the denominator
Step 2: Set the factors equal to zero
 and 
= 0
Step 3: Solve for x
x = -3 and x = 
This should match what you see on the graph of the equation.
The graph of the equation looks like:
Notice that the graph never reaches the two vertical asymptotes.
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Example 2
It does not matter how complex the numerator to the problem is,
the steps for finding vertical asymptotes remain the same. Here
is an example:
Step 1: Factor the denominator
Step 2: Set the factors equal to zero:
 and 
Step 3: Solve for x
x = -4 and x = 1
T hese are the asymptotes.
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