document.write( "Question 975202: Identify the points of discontinuity,holes,vertical asymptotes, and horizontal asmyptotes of each. \r
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Algebra.Com's Answer #597056 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\"%22f%28x%29%22=%28x%5E2%2Bx%29%2F%283x%5E2-9x%29\"
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document.write( "I'll just do the first one. \r\n" );
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document.write( "Factor the numerator and denominator:\r\n" );
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document.write( "Setting the denominator = 0,  3x(x-3) = 0, tells us that\r\n" );
document.write( "we have discontinuities at x=0, and at x=3\r\n" );
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document.write( "We must decide which type of discontinuity we have at 0 and 3.\r\n" );
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document.write( "1. If we can cancel a common factor in the numerator and denominator\r\n" );
document.write( "and remove the discontinuity, then it is a \"removable discontinuity\" \r\n" );
document.write( "or \"a hole in the graph\".\r\n" );
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document.write( "2. If we can't cancel a factor, then the discontinuity is infinite and \r\n" );
document.write( "there is an asymptote there.\r\n" );
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document.write( "We have one of each type.\r\n" );
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document.write( "1. We can remove the discontinuity at x=0 by cancelling the x in the \r\n" );
document.write( "numerator and denominator.  To find out where the hole is, we cancel\r\n" );
document.write( "the x and get a new function which is like the original function everywhere\r\n" );
document.write( "except at the hole.  Let's call it g(x):\r\n" );
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document.write( "\"%22g%28x%29%22=%28x%2B1%29%2F%283%28x-3%29%29\"\r\n" );
document.write( "\"%22g%28x%29%22=%28x%2B1%29%2F%283x-9%29%29\"\r\n" );
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document.write( "g(x) doesn't have a hole at x=0, we substitute and find \r\n" );
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document.write( "\"%22g%280%29%22=%280%2B1%29%2F%283%280%29-9%29=1%2F%28-9%29%29=-1%2F9\"\r\n" );
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document.write( "So f(x) has a hole at \"%28matrix%281%2C3%2C0%2C%22%2C%22%2C-1%2F9%29%29\"\r\n" );
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document.write( "2. We cannot remove the discontinuity at x=3 by cancelling, so there is\r\n" );
document.write( "a vertical asymptote there, a \"non-removable\" discontinuity.\r\n" );
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document.write( "Since the degree of the numerator and denominator have the same degree, 1,\r\n" );
document.write( "there is a horizontal asymptote at y = the ratio of the two leading\r\n" );
document.write( "coefficients, so the horizontal asymptote has equation y=1/3,\r\n" );
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document.write( "So we plot the hole at \"%28matrix%281%2C3%2C0%2C%22%2C%22%2C-1%2F9%29%29\", the vertical asymptote at x=3, and the horizontal \r\n" );
document.write( "asymptote at y=1/3:\r\n" );
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document.write( "Edwin
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