document.write( "Question 1163607: oblique asymptote:
\n" ); document.write( "If a given rational function of higher degree (say 6) does not have a horizontal asymptote, to check for the oblique asymptote...
\n" ); document.write( " ...Do you just keep doing polymomial division repeatedly till you come down to the quotient that is of form mx+b ?
\n" ); document.write( "Thank you \n" ); document.write( "
Algebra.Com's Answer #787742 by ikleyn(52803)\"\" \"About 
You can put this solution on YOUR website!
.
\n" ); document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "A rational function is a ratio of two polynomials\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    f(x) = \"p%28x%29%2Fq%28x%29\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Oblique asymptote is a straight line of the form  y = ax + b.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Oblique asymptote arises when the degree of the polynomial  p(x)  in the numerator is 1 unit greater \r\n" );
document.write( "than the degree of the polynomial  q(x)  in the denominator.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "To get the equation of the oblique asymptote under appropriate condition, you divide the polynomial in the numerator\r\n" );
document.write( "by the polynomial in the denominator (long division).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then you get a linear binomial function as a quotient, and it is the desired equation of the oblique asymptote.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Example:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    f(x) = \"%28x%5E6+-+2%29%2F%28x%5E5+%2B+x%5E4+%2B+x%5E3+%2B+x%5E2+%2B+x+%2B+1%29\" \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The numerator is  p(x) = \"x%5E6+-+2%29\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The denominator is  q(x) = \"%28x%5E5+%2B+x%5E4+%2B+x%5E3+%2B+x%5E2+%2B+x+%2B+1%29\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\"p%28x%29%2Fq%28x%29\" = x - 1 - \"1%2F%28x%5E5+%2B+x%5E4+%2B+x%5E3+%2B+x%5E2+%2B+x+%2B+1%29\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The quotient is  x-1,  and  the equation of the oblique asymptote is  y = x-1  in this case.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "See the plot below.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Plot  y = \"%28x%5E6+-+2%29%2F%28x%5E5+%2B+x%5E4+%2B+x%5E3+%2B+x%5E2+%2B+x+%2B+1%29\" (the given rational function, red) and y = x-1 (oblique asymptote, green)\r\n" );
document.write( "
\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "--------------
\n" ); document.write( "Comment from student: What if the p(x) is degree 8 & q( x) is of degree 3
\n" ); document.write( "--------------\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "My response :\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        Then there is  NO  oblique asymptote.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "        There is a   \"curvilinear asymptote\",  instead.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "See this Wikipedia article\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "https://en.wikipedia.org/wiki/Asymptote#Oblique_asymptotes\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "/\/\/\/\/\/\/\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "It is good that you ask questions.   I like it . . . \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Do not forget to post your  \"THANKS\"  for my teaching.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "/ / / / / / / / / / / / \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "I got next your \"comment\" post with several other questions.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The answers to all these questions are in that Wikipedia article I referred to you.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "You will find answers to all your questions in that article.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "I don't want to re-tell it here, in my post.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Wikipedia makes it better and in more authoritative way . . . \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Happy learning (!)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );