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document.write( "limit (((ln 2))^(x))/(cos ((pi)/4 sin (3 x))) as x \[LongRightArrow] - \[Infinity] = 0 ( T or F )
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document.write( "I placed (copy-pasted) the given expression into the Math editor, and this Math editor interpreted the expression this way\r\n" );
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document.write( " 
.\r\n" );
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document.write( "It is what I will use in my following analysis.\r\n" );
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document.write( " Step 1.\r\n" );
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document.write( "Consider a sequence of real numbers 
, 
, 
, . . . , 
, . . . which approaches to 
from the left.\r\n" );
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document.write( "Then the sequence of real numbers 
, 
, 
, . . . , 
, . . . approaches to 
from the left.\r\n" );
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document.write( "Then the sequence of real numbers 
, 
, 
, . . . , 
approaches to 
= 1/2 from the left.\r\n" );
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document.write( "Then the sequence of real numbers 
, 
, 
, . . . , 
, . . . \r\n" );
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document.write( " approaches to 
= 
= 
from the left.\r\n" );
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document.write( "Then the sequence of numbers 
approaches to 0 (zero) from the right.\r\n" );
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document.write( "Thus for this sequence of numbers 
, the numerator approaches to (ln(2))^(pi/18), which is some constant,\r\n" );
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document.write( "while the denominator 
approaches to zero.\r\n" );
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document.write( "Hence, as the argument x approaches to from the left, \r\n" );
document.write( "the values of the given expression go to positive infinity.\r\n" );
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document.write( "After that, as the argument x becomes greater than , the expression takes finite values, again.\r\n" );
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document.write( " Step 2.\r\n" );
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document.write( "The function in the denominator is periodical with the period 
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document.write( "It means that the behavior of the given expression, which we detected in part (a), repeats inside each period [
,
],\r\n" );
document.write( "for all positive or negative integer numbers \"n\".\r\n" );
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document.write( "In other words, inside each such period, there is a converging sequence of real numbers, \r\n" );
document.write( "for which our expression goes to positive infinity, and after that the expression takes finite values, again.\r\n" );
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document.write( " Step 3.\r\n" );
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document.write( "It means that as x goes to plus or minus infinity, the given expression HAS NO limit.\r\n" );
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document.write( "ANSWER. Of the two possible options, T or F, only F is valid.\r\n" );
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document.write( "Solved.\r
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document.write( "It can be solved in other way, too.\r
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document.write( "In my solution above, I considered a sequence of numbers , converging to pi/6 from the left,\r\n" );
document.write( "and shoved that for such sequence of arguments the whole function has positive values (going to + infinity).\r\n" );
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document.write( "Similarly, if we consider another sequence of numbers , converging to pi/6 from the right,\r\n" );
document.write( "we will get the sequence of the values of the whole function, which all are negative and go to - infinity.\r\n" );
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document.write( "Such a behavior repeats at each interval [,], for all positive or negative integer numbers \"n\".\r\n" );
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document.write( "But the function with such a behavior can not have limits as x goes to +infinity or -infinity.\r\n" );
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document.write( "So, the given function has no limits as x goes to +infinity or -infinity.\r\n" );
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document.write( "![\"\"](\"/images/stars/stars-13.gif\")
