|Module 5 - Limits and Infinity|
|Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Lesson 4 | Self-Test|
|Lesson 5.3: Oblique Asymptotes|
In lesson 5.1 you found vertical and horizontal asymptotes of a rational function. In this lesson you will see an
There appears to be a vertical asymptote at x = -2, which corresponds to the zero of the denominator in the rational function. However there does not seem to be a horizontal asymptote.
5.3.1 Verify that there is no horizontal asymptote by evaluating and . Click here for the answer.
However, there is an important feature of the rational function as x approaches infinity that is not obvious from simply evaluating the limits.
In order to see this feature,
The graph appears to be a line with a small wiggle near the y-axis. This slanted line is an oblique asymptote for the rational function .
Finding the Oblique Asymptote
You can find the equation of an oblique asymptote by converting the rational function to a polynomial plus a
To write as a polynomial plus a proper fraction,
This command converts a rational expression to a polynomial plus a proper fraction.
The result is the sum of a proper fraction and a linear polynomial function x 4. The linear function y = x 4 is the equation of the oblique asymptote.
The Wide View
Graphical support that y = x 4 is an oblique asymptote is provided by graphing both the line and the rational function in a [-100, 100] x [-100, 100] window.
The graphs of the rational function and the line appear to coincide in this window.
The Graphs Differ
To see how the graphs differ,
Notice that the appearance of the graph of a rational function is dramatically affected by the choice of Window values. In a large viewing window the graph of the rational function looks like the graph of the line y = x 4, but in a smaller window, the graphs are not similar.
The Narrow View
Now compare the graphs of the original rational function and the proper fraction .
Near the vertical asymptote at x = 2 the graph of the rational function looks like the graph of y = , the proper fraction.
5.3.2 Discuss the graph of then verify your answer by displaying the graph in a large window and in a window near x = 2. Click here for the answer.
|< Back | Next >|
2007 All rights reserved. |