document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=332814>Question 332814</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help finding a vertical asymptote. I have tried to understand, but it is like another language to me. The problem states,<BR>\n" );
document.write( "FIND THE VERTICAL ASYMPTOTES, IF ANY, OF THE GRAPH OF RATIONAL FUNCTIONS<BR>\n" );
document.write( "X+3/X(X+1)\r<BR>\n" );
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document.write( "I sincerely appreciate any help I can get with this. Thanks! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #238457 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=238457\">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( "A rational function:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ r(x)\ =\ \frac{f(x)}{g(x)}\">\r<BR>\n" );
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document.write( "has a vertical asymptote at any value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> such that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large g(x)\ =\ 0\">, so long as any factors common to both <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f(x)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large g(x)\"> have been eliminated.\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large g(\alpha)\ =\ 0\">, then the line represented by the equation <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ \alpha\"> is a vertical asymptote.\r<BR>\n" );
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document.write( "For your given function, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f(x)\ =\ x\ +\ 3\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large g(x)\ =\ x(x\ +\ 1)\">.  Clearly these two functions have no factors in common, so simply set:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ g(x)\ =\ x(x\ +\ 1)\ =\ 0\">\r<BR>\n" );
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document.write( "And solve for all values of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">.  Each of these values is an <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \alpha\"> value that completes an equation of a vertical asymptote, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ \alpha\"> for your rational function.\r<BR>\n" );
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document.write( "Compare your results to the graph below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28%0D%0A500%2C+500%2C+-5%2C+5%2C+-12%2C+5%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28%0D%0A500%2C+500%2C+-5%2C+5%2C+-12%2C+5%2C%0D%0A%28x%2B3%29%2F%28x%5E2%2Bx%29%0D%0A%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \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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