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