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

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

##### 1 Answer

The limit does not exist...

#### Explanation:

First consider:

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

The various constituent trigonometric functions have periods:

#(2pi)/3, pi/6, (2pi)/7, pi#

The least common multiple of these is

When

If we take a small interval around

Since

Next note that

Note also that all the trigonometric functions are continuous on their various domains.

Hence:

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

is unbounded and takes every value in