document.write( "Question 880407: Does the function f(x)= (x^2+3)/(x^2-1) have a vertical or horizontal asymptote? If so find the equation of the asymptote(s). If not, explicitly state that there are none. \n" ); document.write( "
Algebra.Com's Answer #531403 by KMST(5328)\"\" \"About 
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\"f%28x%29=%28x%5E2%2B3%29%2F%28x%5E2-1%29\" can be transformed into equivalent expressions that will give us the answer.
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\n" ); document.write( "VERTICAL ASYMPTOTES:
\n" ); document.write( "We need to look at the denominator.
\n" ); document.write( "\"f%28x%29=%28x%5E2%2B3%29%2F%28x%5E2-1%29=%28x%5E2%2B3%29%2F%28%28x%2B1%29%28x-1%29%29\" tells us that
\n" ); document.write( "when
\n" ); document.write( "the function is undefined, and as \"x\" approaches those values \"f%28x%29\" increases without bounds.
\n" ); document.write( "We can say \"f%28x%29\" approaches \"infinity\" if your instructor allows that wording,
\n" ); document.write( "or \"lim%28x-%3E-1%2Cf%28x%29%29=infinity\" and \"lim%28x-%3E1%2Cf%28x%29%29=infinity\" if those are the symbols used in your class.
\n" ); document.write( "In any case \"highlight%28system%28x=-1%2C%22and%22%2Cx=1%29%29\" are the vertical asymptotes.
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\n" ); document.write( "HORIZONTAL ASYMPTOTES:
\n" ); document.write( "We need to divide. (Or just look at polynomial degrees and leading coefficients).
\n" ); document.write( " tells us that
\n" ); document.write( "as \"abs%28x%29\" and \"x%5E2\" increase without bounds, \"f%28x%29\" approaches \"1\" ,
\n" ); document.write( "because the term \"4%2F%28x%5E2-1%29\" approaches zero.
\n" ); document.write( "So \"highlight%28y=1%29\" is the horizontal asymptote.
\n" ); document.write( "(You could tell that the rational function \"f%28x%29=%28x%5E2%2B3%29%2F%28x%5E2-1%29\" would have a horizontal asymptote, because the degree of numerator \"x%5E2%2B3\" is not greater than the degree of denominator \"x%5E2-1\" . You could tell that the asymptote would be \"y=1\" because the degrees of numerator and denominator are the same and the ratio of the leading coefficients of numerator and denominator is \"1%2F1=highlight%281%29\" ).
\n" ); document.write( "The graph of \"red%28f%28x%29%29\" and its asymptotes is \n" ); document.write( "
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