How do I find the limit as x approaches infinity of a trigonometric function?

#lim_(x->oo)(x^2csc3xtan6x)/(cos7xcot^2x)#

Answer 1

The limit does not exist...

First, think about:

#f(x) = (csc 3x tan 6x)/(cos 7x cot^2 x)#

There are periods for each of the component trigonometric functions:

#(2pi)/3, pi/6, (2pi)/7, pi#
The least common multiple of these is #2pi#. Hence #f(x)# has period #2pi# or a factor thereof. In fact we can tell that it has period exactly #2pi# since #7# is prime.
When #x=pi/14# we find that #cos 7x = cos (pi/2) = 0# and all of the other trigonometric functions are non-zero.
If we take a small interval around #x = pi/14# then #cos 7x# changes sign while the other trigonometric functions retain their signs. Hence #f(x)# changes sign in a small interval around #x = pi/14#.
Since #cos 7x# is in the denominator, this means that #f(x)# has a vertical asymptote with different signs on either side of the asymptote at #x=pi/14 + 2npi# for any integer #n#.
Next note that #x^2->oo# as #x->oo# (or #x->-oo#).

Additionally, keep in mind that every trigonometric function is continuous across all of its domains.

Hence:

#(x^2 csc 3x tan 6x)/(cos 7x cot^2 x)#
is unbounded and takes every value in #(-oo, oo)# repeatedly as #x->oo# (or #x->-oo#). It definitely does not converge to a limit.
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Answer 2

To find the limit as ( x ) approaches infinity of a trigonometric function, consider the behavior of the function as ( x ) becomes increasingly large. Identify any periodic behavior and determine if the function approaches a finite value or oscillates indefinitely. If the function contains terms like sine or cosine, consider using trigonometric identities to simplify the expression before evaluating the limit. If the limit exists and is finite, that value is the limit as ( x ) approaches infinity. If the limit is infinite or oscillates indefinitely, state that the limit does not exist.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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