L1.3 A1 Rational Functions - Part 4
Completion requirements
Unit 1
Functions
Horizontal Asymptotes
You may have noticed the graphs of many rational functions also have horizontal asymptotes. These can be most easily determined by comparing the highest-degree terms of the numerator and denominator.If the degree of the numerator is greater than the degree of the denominator, such as in, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«mi»x«/mi»«mo»-«/mo»«mn»7«/mn»«/mrow»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math» the numerator will grow faster than the denominator as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» becomes a large positive or negative value, and there will be no horizontal asymptote.
If the degree of the numerator is less than the degree of the denominator,
such as in «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mfrac»«mrow»«mi»x«/mi»«mo»-«/mo»«mn»4«/mn»«/mrow»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»7«/mn»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«/mfrac»«/mstyle»«/math»,
the denominator will grow faster than the numerator, and the value of
«math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math»
will approach zero for large positive and negative «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math»-values. This
results in a horizontal asymptote of «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math».
