document.write( "Question 147070: Find the vertical and horizontal asymptote(s)\r
\n" ); document.write( "\n" ); document.write( "f(x)=(x^3+3x^2)/(x^2-4)
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Algebra.Com's Answer #107425 by solver91311(24713) $\"\"$ $\"About$

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A rational function has vertical asymptotes with equation $\"x=a\"$ for any real number value of $\"a\"$ that makes the denominator equal zero. So, take the denominator polynomial, set it equal to zero, and solve. Any real zeros of that equation will be the values of a in $\"x=a\"$ to describe your vertical asymptotes. The given problem will be a little easier if you recognize that the denominator is the difference of two squares.\r
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\n" ); document.write( "\n" ); document.write( "To find horizontal asymptotes, first compare the degree of the denominator polynomial to the degree of the numerator polynomial.\r
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\n" ); document.write( "\n" ); document.write( "If the degree of the denominator is greater than the degree of the numerator, there is a horizontal asymptote at $\"y=0\"$ \r
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\n" ); document.write( "\n" ); document.write( "If the degree of the denominator and the degree of the numerator are equal, there is a horizontal asymptote at $\"y=p%2Fq\"$ where p is the lead coefficient on the numerator and q is the lead coefficient on the denominator.\r
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\n" ); document.write( "\n" ); document.write( "If the degree of the denominator is less than the degree of the numerator, then there is no horizontal asymptote. In this case, if the degrees differ by 1, then there is a straight line slant (or oblique) asymptote whose equation is the quotient part (excluding the remainder) of the polynomial long division of the denominator into the numerator. This is the case with your example. \n" ); document.write( "

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