document.write( "Question 123152This question is from textbook college algebra/ blitzer
\n" ); document.write( ": Please help solve
\n" ); document.write( "Find the equations of the horizontal and vertical asymptotes for the following. Type none if the function does not have an asympote. \r
\n" ); document.write( "\n" ); document.write( "a) f(X)=2x+3/x+2
\n" ); document.write( "Answer: Horizontal
\n" ); document.write( "Vertical\r
\n" ); document.write( "\n" ); document.write( "B) g(x)=5x/x^2+1
\n" ); document.write( "Answer
\n" ); document.write( "Horizontal
\n" ); document.write( "Vertical \n" ); document.write( "
Algebra.Com's Answer #90409 by solver91311(24713)\"\" \"About 
You can put this solution on YOUR website!

\n" );
document.write( "

\n" );
document.write( "Discussion
\n" );
document.write( "

\n" );
document.write( "

\n" );
document.write( "Horizontal Asymptotes\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "There is never more than one horizontal asymptote.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "Compare the degree of the numerator polynomial to the degree of the denominator
\n" );
document.write( "polynomial.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "If the degree of the denominator is larger than the degree of the numerator,
\n" );
document.write( "then the horizontal asymptote is always \"y=0\".  In other words, the 
\n" );
document.write( "function is asymptotic to the x-axis.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "If the degree of the denominator is smaller than the degree of the numerator,
\n" );
document.write( "then there is no horizontal asymptote.  There is, however, a slant asymptote.
\n" );
document.write( "To determine the slant asymptote, use polynomial long division.  The equation
\n" );
document.write( "of the slant asymptote is equal to the quotient excluding the remainder.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "If the degree of the denominator equals the degree of the numerator, the
\n" );
document.write( "horizontal asymptote is at y = a where a is the quotient of the lead
\n" );
document.write( "coefficient on the numerator divided by the lead coefficient on the
\n" );
document.write( "denominator.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "Vertical asymptotes\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "There is one vertical asymptote for every real zero of the denominator
\n" );
document.write( "polynomial.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "Set the denominator polynomial equal to zero and solve for all real roots.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "The vertical asymptotes are then \"x=a%5B1%5D\", \"x=a%5B2%5D\", ..., \"x=a%5Bn%5D\"\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "Where \"a%5B1%5D\" through \"a%5Bn%5D\" are all of the real roots of the 
\n" );
document.write( "denominator polynomial.
\n" );
document.write( "

\n" );
document.write( "

\n" );
document.write( "

\n" );
document.write( "Solution
\n" );
document.write( "

\n" );
document.write( "

\n" );
document.write( "I'm going to show a solution for your first problem, then you should be able
\n" );
document.write( "to solve the other one.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "\"f%28x%29=%282x%2B3%29%2F%28x%2B2%29\"\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "Horizontal:  The degree of the numerator is 1.  The degree of the denominator
\n" );
document.write( "is 1.  The lead coefficient on the numerator is 2 and the lead coefficient on
\n" );
document.write( "the denominator is 1.  \"2%2F1=2\"\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "Therefore, the horizontal asymptote is \"y=2\"\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "Vertical asymptote:  Set the denominator equal to zero and solve.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "\"x%2B2=0\"
\n" );
document.write( "\"x=-2\"\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "Therefore the vertical asymptote is \"x=-2\".  Since the denominator
\n" );
document.write( "polynomial is of degree 1, we know that there is only one real zero of the
\n" );
document.write( "polynomial and therefore only one vertical asymptote.
\n" );
document.write( "

\n" );
document.write( "

\n" );
document.write( "

\n" );
document.write( "Check Answer
\n" );
document.write( "

\n" );
document.write( "

\n" );
document.write( "\"graph%28600%2C600%2C-5%2C5%2C-5%2C5%2C%282x%2B3%29%2F%28x%2B2%29%29\"\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "The graphing system on this site doesn't handle functions with vertical
\n" );
document.write( "asymptotes very well, so ignore the sort of vertical line on the graph
\n" );
document.write( "above.  Note that there is a vertical line at \"x=-2\" that the graph
\n" );
document.write( "does not intersect, and a horizontal line at \"y=2\" that is approached
\n" );
document.write( "by the function as the absolute value of x gets very large.
\n" );
document.write( "
\r
\n" );
document.write( "\n" );
document.write( " \n" );
document.write( "
\n" );