Algebra.Com's Answer #851770 by ikleyn(52781)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
.
\n" );
document.write( "Find : limit  , when (x→3)
\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "\r\n" );
document.write( "The floor function rounds a real number x down to the nearest integer \r\n" );
document.write( "less than or equal to it. In other words, it returns the largest integer that is not greater than x.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " Notice that the given function is NOT defined at x = 3.\r\n" );
document.write( "\r\n" );
document.write( " Due to this reason, the limit of the given function\r\n" );
document.write( " can not be determined directly.\r\n" );
document.write( "\r\n" );
document.write( " Let's consider this issue from different points of view.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "When x approaches to 3 from the right side,  approaches to 1 from the right side.\r\n" );
document.write( "\r\n" );
document.write( "Therefore,  has then the value of 1 identically in vicinity of x= 3 on its right side.\r\n" );
document.write( " \r\n" );
document.write( "Hence, function  in the numerator is then identically 0 (zero) in vicinity of x = 3 on its right side.\r\n" );
document.write( "\r\n" );
document.write( "Therefore, the right limit of  as x ---> 3 from the right side can be naturally defined as 0 (zero).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "On the contrary, when x approaches to 3 from the left side, approaches to 1 from the left side.\r\n" );
document.write( "\r\n" );
document.write( "Hence, function  is then identically -1 in vicinity of x= 3 on its left side.\r\n" );
document.write( "\r\n" );
document.write( "Therefore, the left limit of as x ---> 3 does not exist (is positive infinity).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus, as x ---> 3, the limit of the function in the common sense does not exist.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER. The given function is not defined at x = 3.\r\n" );
document.write( "\r\n" );
document.write( " The right limit of the given function at x --> 3 is (or can be defined as) 0 (zero); \r\n" );
document.write( "\r\n" );
document.write( " the left limit of the given function at x --> 3 does not exist;\r\n" );
document.write( "\r\n" );
document.write( " the limit of the given function at x --> 3 in common sense does not exist and can not be defined.\r\n" );
document.write( " \r
\n" );
document.write( "\n" );
document.write( "Solved.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "What I placed in this my post, is a standard analysis and a standard mantra
\n" );
document.write( "to pronounce when solving such problems.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( " \n" );
document.write( " |