Question 1180227
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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.
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<pre>
F A L S E.



                 Counter-example:



Let  a(x) = {{{1/x}}} at x =/= 0  and  a(0) = 0  

            (not bounded and not continuous function on R,
             but defined over entire R).



Let  b(x) = {{{1/(x-1)}}} at x =/= 1  and  b(1) = 0  

            (not bounded and not continuous function on R
             but defined over entire R).




The limit of  a(x)*b(x)  at x -->  -oo  does exist and is equal to 0 (zero).


The limit of  a(x)*b(x)  at x -->  oo  does exist and is equal to 0 (zero).
</pre>

Solved, &nbsp;answered and explained.



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Probably, &nbsp;the answer would be different, &nbsp;had the problem require functions &nbsp;a(x) &nbsp;and &nbsp;b(x) &nbsp;be continuous;


but in the given post, &nbsp;there is &nbsp;NO &nbsp;such a requirement, &nbsp;so I used this fact and constructed counter-example with 

discontinuous functions.