document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=202900>Question 202900</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Which one of the following is true? (1) The function, f(x)=2x^2+1/x^2-1, does not have a horizontal asymptote. (2) The graph of a rational function cannot have x-intercepts. (3) The horizontal asymptote for the graph of y=4x-1/x+3 is x=-3. (4) The graph of f(x)=2x+4/x^2-4 has only one vertical asymptote.  THIS IS A TOUGH ONE....HELP. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #153063 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=153063\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "1) Has a horizontal asymptote at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y = 2\"> because the numerator and denominator are of the same degree, hence there is a horizontal asymptote at the value of the lead coefficient of the numerator divided by the lead coefficient of the denominator.  This one is false.\r<BR>\n" );
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document.write( "2) The graph of a rational function can have <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> intercepts.  The graph will intercept the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-axis at any zero of the numerator polynomial that is not also a zero of the denominator polynomial.  Consider <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{x^2-3x+2}{4 - x}\">.  This one is false.\r<BR>\n" );
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document.write( "3) As in #1, the horizontal asymptote is at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y = a\"> where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\"> is the value of the lead coefficient of the numerator divided by the lead coefficient of the denominator.  Hence, the horizontal asymptote is at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y = 4\"> not <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y = -3\">.  This one is false.\r<BR>\n" );
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document.write( "4) This one looks false on the face of it, because you would expect a vertical asymptote at any zero of the denominator polynomial.  However, note that -2 is a zero of the denominator, but it is also a zero of the numerator.  Hence, while there is a discontinuity at -2, there is no asymptote.  In order to prove it, you need a little trick from the Calculus called L'Hôpital's Rule.  It says that if:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\rightarrow c}\ f(x) = 0\text{ or }\pm\infty\">\r<BR>\n" );
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document.write( "and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\rightarrow c}\ g(x)= 0\text{ or }\pm\infty\">\r<BR>\n" );
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document.write( "and if\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\rightarrow c}\ \frac{f'(x)}{g'(x)}\"> exists, then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\rightarrow c}\ \frac{f(x)}{g(x)} = \lim_{x\rightarrow c}\ \frac{f'(x)}{g'(x)}\">\r<BR>\n" );
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document.write( "So for the function given in #4,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ f'(x) = 2\"> and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ g'(x) = 2x\"> and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\rightarrow -2}\ \frac{f'(x)}{g'(x)} = \frac{2}{2(-2)}=-\frac{1}{2}\">\r<BR>\n" );
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document.write( "not <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \pm\infty\"> as would be the case if there were an asymptote at that point.\r<BR>\n" );
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document.write( "Therefore #4 is the true statement.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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