document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1090612>Question 1090612</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let f(x) be a rational function such that when f(x) is in reduced form, the numerator and denominator both have degree 3. What sorts of asymptotes could f(x) have?<BR>\n" );
document.write( "a. Three vertical asymptotes only<BR>\n" );
document.write( "b. One vertical asymptote and one oblique asymptote<BR>\n" );
document.write( "c. One horizontal asymptote and at least one vertical asymptote<BR>\n" );
document.write( "d. One horizontal and one oblique asymptote<BR>\n" );
document.write( "e. No vertical asymptotes and no horizontal asymptotes </SPAN>\n" );
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document.write( "The graph of  f(x) will have vertical asymptotes at those values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > for which the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=denominator ALIGN=MIDDLE  ALT=\"denominator\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equal to zero.=> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=denominator ALIGN=MIDDLE  ALT=\"denominator\"   DISABLED_style= \"max-width:90%; height:auto;\" > could be equal to zero for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=one ALIGN=MIDDLE  ALT=\"one\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=two ALIGN=MIDDLE  ALT=\"two\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=three ALIGN=MIDDLE  ALT=\"three\"   DISABLED_style= \"max-width:90%; height:auto;\" > values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=n ALIGN=MIDDLE  ALT=\"m=n\"   DISABLED_style= \"max-width:90%; height:auto;\" > (that is, the degrees of the numerator and denominator are the same), then the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > will have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=one ALIGN=MIDDLE  ALT=\"one\"   DISABLED_style= \"max-width:90%; height:auto;\" > horizontal asymptote at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%5Bn%5D%2Fb%5Bm%5D ALIGN=MIDDLE  ALT=\"y=a%5Bn%5D%2Fb%5Bm%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >. (coefficients of highest degree)\r<BR>\n" );
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document.write( "When the degree of the numerator is exactly <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=one ALIGN=MIDDLE  ALT=\"one\"   DISABLED_style= \"max-width:90%; height:auto;\" > more than the degree of the denominator, the graph of the rational function will have an oblique asymptote.\r<BR>\n" );
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document.write( "in your case, the degree of the numerator and denominator are same; so, there is no oblique asymptote  \r<BR>\n" );
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document.write( "so, your answer is:\r<BR>\n" );
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document.write( "c. One horizontal asymptote and at least one vertical asymptote\r<BR>\n" );
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document.write( " </SPAN>\n" );
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