document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163632>Question 1163632</A>: <I><SPAN STYLE=\"word-break: break-all;\"> While graphing rational functions, when there are 2 vertical asymptotes and a horizontal symptote (giving 6 sectors), one of the three segments of the resulting graph seem to go through the horizontal asymptote (some times once and at other times twice). And no segment of the graph seems to cross the vertical asymptote, however.\r<BR>\n" );
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document.write( "Question: So why do we have horizontal asymptotes that seem to be not an absolute  barrier or are there any conditions in general when can you expect a segment of the graph cross the horizontal asymptote.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #787783 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=787783\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Horizontal asymptotes describe the behavior of the function when the value of the independent variable either increases or decreases without bound.  So the fact that the function is asymptotic to a given horizontal line does not mean that the value of the function cannot assume the function value of the horizontal line, it just means that the function will have that horizontal line function value as a limit whenever x gets really large in either direction.\r<BR>\n" );
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document.write( "So graph your function and find the smallest (most negative value) where the function intersects the line <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ \frac{a}{b}\"> where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\"> is the lead coefficient on the numerator and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\"> is the lead coefficient on the denominator (for numerator and denominator of equal degree) or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ 0\"> (for numerator of lesser degree than the denominator) and find the largest on the positive side, and then say, as far as horizontal asymptotes go, I don't care about the behavior of the function in between those two points, it is only on the ends that it matters.\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" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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