(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I am trying to find the limit of the following question:

lim (as x approaches infinite) sin[ arctan(((x^3)+2)/((x^2)+5)) + (((cosx)^2)/((|x|)^(1/2)))*arctan(2x)]

(to see the above function better, without a million brackets, please use the following link)

http://www.wolframalpha.com/input/?i=sin[+arctan%28%28%28x^3%29%2B2%29%2F%28%28x^2%29%2B5%29%29+%2B+%28%28%28cosx%29^2%29%2F%28%28|x|%29^%281%2F2%29%29%29*arctan%282x%29]

2. Relevant equations

3. The attempt at a solution

My attempt so far is practically non-existent! This question is part of the harder set of exercises available from my university's department for first year calc, but I figure that if I can get to a point where I can solve these kinds of problems, I should have no problem on the test!

OK, so before I even start describing my attempt, I have a quick question. Say we were given something like this:

Find lim as x approaches infinity of sin(cos(f(x))), where f(x) is of the form 0/0 or infinity over infinity, is it correct to simply isolate f(x), use hoptals rule until you come to a conclusion, and then plug it back into the limit?

Also, since the orignial question is the limit as x approaches POSITIVE infinity, can we just conclude that the |x|=x?

OK so back to the attempt, if we truly are aloud to use Hopitals rule inside of trig functions, then I get the first arctan(f(x)) to be equal to pi/2. Is this correct so far?

As for the second term within the sin, I cannot solve this! I tried making |x|=x and combined it into a single fraction. But I seem to get 0/infinity and I cannot find out how to simplify this so that it is of a form that I can use with Hopital's rule.

I realize that this is a very loaded question, but thank you so much for your help!

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# Finding the limit of a very complicated trig function + some extra questions

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