document.write( "Question 1200106: Find all horizontal and vertical asymptotes of the function: f(x) = x/sqrt(x^2-1)
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Algebra.Com's Answer #834132 by Edwin McCravy(20056) $\"\"$ $\"About$
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document.write( "To find vertical asymptotes, set the denominator equal to 0:\r\n" );
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document.write( "x-1 = 0;  x+1 = 0\r\n" );
document.write( "  x = 1;    x = -1\r\n" );
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document.write( "So the vertical asymptotes are the vertical lines whose equations are\r\n" );
document.write( "x = 1  and x = -1\r\n" );
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document.write( "To find the horizontal asymptotes we substitute large positive and negative\r\n" );
document.write( "numbers for x, and see if they approach any finite number.\r\n" );
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document.write( "Substituting x = 1000,\r\n" );
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document.write( "That is very close to 1, so we assume y = 1 is a horizontal asymptote.\r\n" );
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document.write( "Substituting x = -1000,\r\n" );
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document.write( "That is very close to -1, so we assume y = -1 is also a horizontal asymptote.\r\n" );
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document.write( "So we draw the vertical asymptotes x = 1 and x= -1 and horizontal asymptotes \r\n" );
document.write( "are y = 1 and y = -1\r\n" );
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document.write( "Since the function contains a square root, we must ensure that what's under the\r\n" );
document.write( "square root is greater than 0.\r\n" );
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document.write( "They have zeros 1 and -1.  We make a number line\r\n" );
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document.write( "----------o-----o---------\r\n" );
document.write( "-4 -3 -2 -1  0  1  2  3  4 \r\n" );
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document.write( "Choose test point x=-2 in interval \r\n" );
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document.write( "That is a true inequality, so  is part of\r\n" );
document.write( "the domain.\r\n" );
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document.write( "Choose test point x=0 in interval \r\n" );
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document.write( "That is a false inequality, so  is NOT part of\r\n" );
document.write( "the domain.  Therefore there is no graph between where x=-1 and where x=1.\r\n" );
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document.write( "Choose test point x=2 in interval \r\n" );
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document.write( "That is a true inequality, so  is part of\r\n" );
document.write( "the domain.  So the shaded number line is\r\n" );
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document.write( "<==========o-----o=========>\r\n" );
document.write( " -4 -3 -2 -1  0  1  2  3  4 \r\n" );
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document.write( "So the domain is  \r\n" );
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document.write( "We find a couple points in both parts of the domain, say the points \r\n" );
document.write( "(-2,-1.2) and (2,1.2).  Then sketch the graph:\r\n" );
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document.write( "Edwin
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