document.write( "Question 1180227: True or False, if False provide a counterexample: For functions a & b defined on the entire real line, if both a and b are not bounded on R (Real), then the limit to infinity of the product of a and b cannot exist. \n" ); document.write( "

Algebra.Com's Answer #809937 by ikleyn(52790) $\"\"$ $\"About$

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\n" ); document.write( "True or False, if False provide a counterexample: \r
\n" ); document.write( "\n" ); document.write( "For functions a & b defined on the entire real line, if both a and b are not bounded on R (Real),
\n" ); document.write( "then the limit to infinity of the product of a and b cannot exist.
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document.write( "F A L S E.\r\n" );
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document.write( "                 Counter-example:\r\n" );
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document.write( "Let  a(x) =  at x =/= 0  and  a(0) = 0  \r\n" );
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document.write( "            (not bounded and not continuous function on R,\r\n" );
document.write( "             but defined over entire R).\r\n" );
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document.write( "Let  b(x) =  at x =/= 1  and  b(1) = 0  \r\n" );
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document.write( "            (not bounded and not continuous function on R\r\n" );
document.write( "             but defined over entire R).\r\n" );
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document.write( "The limit of  a(x)*b(x)  at x -->  -oo  does exist and is equal to 0 (zero).\r\n" );
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document.write( "The limit of  a(x)*b(x)  at x -->  oo  does exist and is equal to 0 (zero).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved, answered and explained.\r
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\n" ); document.write( "\n" ); document.write( "Probably, the answer would be different, had the problem require functions a(x) and b(x) be continuous;\r
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\n" ); document.write( "\n" ); document.write( "but in the given post, there is NO such a requirement, so I used this fact and constructed counter-example with \r
\n" ); document.write( "\n" ); document.write( "discontinuous functions.\r
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