document.write( "Question 489623: Which function dominates as x approaches infinity ?
\n" ); document.write( "5*2^x OR 69000x^18 \n" ); document.write( "
Algebra.Com's Answer #333710 by Edwin McCravy(20056)\"\" \"About 
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document.write( "f(x) is said to dominate over g(x) if \r\n" );
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document.write( "  lim\"%22g%28x%29%22%2F%22f%28x%29%22\" = 0\r\n" );
document.write( " x->\"infinity\"\r\n" );
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document.write( "5*2x dominates because\r\n" );
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document.write( " lim \"%2869000x%5E18%29%2F%285%2A2%5Ex%29\" = 0\r\n" );
document.write( "x->\"infinity\"\r\n" );
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document.write( "The reason we know this is because\r\n" );
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document.write( "1. This is a case of infinity over infinity \"infinity%2Finfinity\", thus\r\n" );
document.write( "   L'Hopital's rule applies.\r\n" );
document.write( "2. Each derivative of the numerator up through the 17th will result\r\n" );
document.write( "   in a non-zero constant times a power of x and will thus approach \r\n" );
document.write( "   infinity.\r\n" );
document.write( "3. Each derivative of the denominator up through the 17th will result in a\r\n" );
document.write( "   constant times 2x and will thus approach infinity.\r\n" );
document.write( "4. Thus up through the 17th derivatives of the numerator and denominator\r\n" );
document.write( "   the result will be a case of \"infinity%2Finfinity%22\" and L'Hopital's rule will apply. \r\n" );
document.write( "5. The 18th derivative of the numerator will result in a constant\r\n" );
document.write( "6. The 18th derivative of the denominator will still result in a constant\r\n" );
document.write( "   times 2x which is a case of a constant over infinity\r\n" );
document.write( "7. The limit of a constant over an expression that approaches infinity\r\n" );
document.write( "   as x approaches infinity is zero. \r\n" );
document.write( "   \r\n" );
document.write( " lim \"%2869000x%5E18%29%2F%285%2A2%5Ex%29\" = \r\n" );
document.write( "x->\"infinity\"\r\n" );
document.write( " \r\n" );
document.write( " lim \"%2813800x%5E18%29%2F%282%5Ex%29\" =\r\n" );
document.write( "x->\"infinity\"\r\n" );
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document.write( "13800 lim \"%28x%5E18%29%2F%282%5Ex%29\" =\r\n" );
document.write( "     x->\"infinity\"\r\n" );
document.write( "... =\r\n" );
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document.write( "\"13800%2A18%21%2F%28ln%282%29%29%5E18\"lim \"1%2F%282%5Ex%29\" = 0\r\n" );
document.write( "      x->\"infinity\"\r\n" );
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document.write( "Edwin
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