document.write( "Question 42098: Determine the values of x for any holes in the graph of:\r
\n" ); document.write( "\n" ); document.write( "f(x)= (x+3)/x^2+5x+6. \n" ); document.write( "
Algebra.Com's Answer #27260 by AnlytcPhil(1806)\"\" \"About 
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Determine the values of x for any holes in the graph of: \r\n" );
document.write( "f(x)= (x+3)/x^2+5x+6.\r\n" );
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document.write( "A hole occurs in a rational function f(x) at x=a if\r\n" );
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document.write( "f(a) is not defined at x=a due to numerator and denominator\r\n" );
document.write( "becoming 0, and\r\n" );
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document.write( "lim f(x) = lim f(x) = lim f(x) = some finite number\r\n" );
document.write( "x->a-      x->a+      x->a \r\n" );
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document.write( "f(x) = (x+3)/[(x+3)(x+2)]  \r\n" );
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document.write( "We cannot cancel (x+3)'s except when x is not equal to -3,\r\n" );
document.write( "for f(x) becomes the meaningless \"0/0\" and is not defined \r\n" );
document.write( "when x=-3.\r\n" );
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document.write( "However for every other number besides x=-3, \r\n" );
document.write( "function f(x) is identical with g(x) = 1/(x+2) because\r\n" );
document.write( "the (x+3)'s can be canceled when (x+3) is not 0.\r\n" );
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document.write( "So\r\n" );
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document.write( "lim f(x) = g(-3) = 1/(-3+2) = -1\r\n" );
document.write( "x->-3\r\n" );
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document.write( "So there is a hole in the curve at x = -3.\r\n" );
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document.write( "A hole in the curve is often called a \"removable discontinuity\"\r\n" );
document.write( "because the hole could be plugged up by defining f(-3) as -1.\r\n" );
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document.write( "Edwin
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