rational.html

Asymptotes (using tables and graphs)

Below is a computer generated graph of a rational function with the coordinate system removed from the picture. We know that a vertical line must intersect the graph of a function at most once. Why does this "function" seems to include a vertical line? By studying properties of rational functions we will be able to answer this question.

Example 1

The function , used in the previous graph, has a domain which includes all real numbers except for 2 which is the value making the denominator equal to zero. Make tables showing what happens to this function as x gets close to 2 from the right and as x gets close to 2 from the left.

Solution .

The table below shows values for the function as x get closer to 2 from the right side (i.e. x values larger than 2 but close to 2).

     x            func(x)

----------------------------

 2.100000000      30.00000000

 2.010000000      300.0000000

 2.001000000      3000.000000

 2.000100000      30000.00000

 2.000010000      300000.0000

 2.000001000      3000000.000

 2.000000100      30000000.00

As x gets closer to 2 from the right, the y values get larger and larger. We say that y approaches infinity.

This table shows values for the function as x gets closer to 2 from the left. Note that the y values are negative but have large absolute values.

     x            func(x)

----------------------------

 1.900000000     -30.00000000

 1.990000000     -300.0000000

 1.999000000     -3000.000000

 1.999900000     -30000.00000

 1.999990000     -300000.0000

 1.999999000     -3000000.000

 1.999999900     -30000000.00

Here as x approaches 2 from the left, y approaches negative infinity. The line x =2 is called a vertical asymptote for the function.

Example 2

Experiment with graphs of this function with x between 0 and 4.

Solution .

Here is the graph of the function with x between 0 and 4.

Note that a vertical line is drawn. This line at x = 2 is not actually part of the function's graph. Since the computer connects points it uses with line segments, this gives the appearance that x = 2 is part of the graph. To see the points used in creating the graph we graph the function with style equal to points (dot mode on your calculator).

If we restrict the range of y values we will see

Here is what we get when we do not incorrectly connect the last dot to the left of 2 with the first do to the right of 2.

In the next example we will examine the behavior of y when x is large.

Example 3

Make a table for this function as x gets larger and larger.

Solution .

In the table below, x starts out as 0.1 and increases by a factor of 100 in each line of the table until it reaches 100000000000.00 ( expressed in scientific notation in the table as 0.1000000000E12). The y values become very small positive numbers.

     x            func(x)

----------------------------

 .1000000000       -1.578947369

 10.00000000        .3750000000

 1000.000000        0.3006012024E-2

 100000.0000        0.3000060000E-4

 10000000.00        0.3000000600E-6

 1000000000.        0.3000000006E-8

 0.1000000000E12    0.3000000000E-10

We say that as x approaches infinity, y approaches 0.

Here is a table with x approaching negative infinity.

     x            func(x)

----------------------------

-.1000000000       -1.428571429

-10.00000000       -.2500000000

-1000.000000       -0.2994011976E-2

-100000.0000       -0.2999940001E-4

-10000000.00       -0.2999999400E-6

-1000000000.       -0.2999999994E-8

-0.1000000000E12   -0.3000000000E-10

We say that as x approaches negative infinity, y approaches 0. The equation y = 0 is called the horizontal asymptote .

If we plot this function again and zoom out a little we will see the y values get close to zero as x gets bigger and bigger.

If we draw the graph again using boxed axes we see

Example 4

Let . Find the vertical asymptotes, use tables to find the horizontal asymptote, and make a graph of this function.

Solution .

The denominator is zero when x = 2 or x = -2. The following tables indicate that y approaches infinity or negative infinity for x near 2 or -2. Thus, x = 2 and x = -2 are vertical asymptotes.

     x            func(x)

----------------------------

 1.900000000     -6.192307692

 1.990000000     -73.68796992

 1.999000000     -748.6875469

 1.999900000     -7498.687505

 1.999990000     -74998.50000

 1.999999000     -749998.5000

 1.999999900     -7499998.500

     x            func(x)

----------------------------

 2.100000000      8.817073171

 2.010000000      76.31296758

 2.001000000      751.3125469

 2.000100000      7501.312505

 2.000010000      75001.50000

 2.000001000      750001.5000

 2.000000100      7500001.500

     x            func(x)

----------------------------

-2.100000000      8.817073171

-2.010000000      76.31296758

-2.001000000      751.3125469

-2.000100000      7501.312505

-2.000010000      75001.50000

-2.000001000      750001.5000

-2.000000100      7500001.500

     x            func(x)

----------------------------

-1.900000000     -6.192307692

-1.990000000     -73.68796992

-1.999000000     -748.6875469

-1.999900000     -7498.687505

-1.999990000     -74998.50000

-1.999999000     -749998.5000

-1.999999900     -7499998.500

To see what happens to the value of the function when x is large look at the following table.

     x            func(x)

----------------------------

 .1000000000        .7481203008

 10.00000000        1.531250000

 1000.000000        1.500003000

 100000.0000        1.500000000

 10000000.00        1.500000000

 1000000000.        1.500000000

 0.1000000000E12    1.500000000

Based on the previous table and the next one,

     x            func(x)

----------------------------

-.1000000000        .7481203008

-10.00000000        1.531250000

-1000.000000        1.500003000

-100000.0000        1.500000000

-10000000.00        1.500000000

-1000000000.        1.500000000

-0.1000000000E12    1.500000000

we say that y = 3/2 is the horizontal asymptote.

Here is a graph of the function.

The purpose of the next example is to show that not every root of the denominator gives a vertical asymptote.

Example 5

Conside the function . Show that it only has one vertical asymptote.

Solution .

The denominator is zero when x = -1 or x = 1. If we make a table for x near -1 we will see that the function does not approach infinity.

     x            func(x)

----------------------------

-1.100000000     -.4761904762

-1.010000000     -.4975124378

-1.001000000     -.4997501249

-1.000100000     -.4999750012

-1.000010000     -.5000000000

-1.000001000     -.5000000000

-1.000000100     -.5000000000

     x            func(x)

----------------------------

-.9000000000     -.5263157895

-.9900000000     -.5025125628

-.9990000000     -.5002501251

-.9999000000     -.5000250013

-.9999900000     -.5000025000

-.9999990000     -.5000000000

-.9999999000     -.5000000000

When x is near -1, y is close to -0.5. Thus, x = -1 is not a vertical asymptote since the function does not approach infinity or negative infinity when x is close to -1.

x =1 is a vertical asymptote as the following tables suggest.

     x            func(x)

----------------------------

 .9000000000     -10.00000000

 .9900000000     -100.0000000

 .9990000000     -1000.000000

 .9999000000     -10000.00000

 .9999900000     -100000.0000

 .9999990000     -999999.5000

 .9999999000     -9999999.500

     x            func(x)

----------------------------

 1.100000000      10.00000000

 1.010000000      100.0000000

 1.001000000      1000.000000

 1.000100000      10000.00000

 1.000010000      100000.5000

 1.000001000      1000000.500

 1.000000100      10000000.50

The fact that x = -1 is not a vertical asymptote can be seen from the original equation. Note that the function factors into which reduces ( as long as x is not -1) to . This curve has only one vertical asymptote at x = 1.

Here is the graph of this function. Note that there should be a missing point at (-1, -0.5) since the original function is not defined when x = -1.