Study functions at $-\infty$, 0 and $+\infty$ The functions are:
$f(x) = \frac{(x-\sin(x))\log(1+x^4)}{x^7}$ 
and 
$g(x) = \frac{x^3-\arctan(x)\log(1+x^2)}{x^5}$ 
I know that $\lim_{x \rightarrow 0} f(x) = \frac{1}{6}$, $\lim_{x \rightarrow 0} g(x) = \frac{5}{6}$ and that $\lim_{x \rightarrow -\infty} f(x) = \lim_{x \rightarrow +\infty} f(x)$, $\lim_{x \rightarrow -\infty} g(x) = \lim_{x \rightarrow +\infty} g(x)$  but I don't know how to evaluate the limits at $+\infty$
These are possible exam questions so, please, don't answer with things like "It's safe to say..." (I need a good justified answer). Also don't mark this as duplicate, as I had to delete a question I made because it was marked as duplicate of another question with similar, but completely different (a-b*c != (a-b)*c) question.
 A: The first function grows $\sim \frac{\log x}{x^6}$ which of course converges to $0$. For the second function use $\arctan x \sim \frac{\pi}{2} - \frac{1}{x}+O(\frac{1}{x^3})$ and see if you get the convergence. 
A: I'm not the fanciest maths type; I humbly offer some ideas bouncing your way.
Hmm, could you try approximating with area under the curve method?
Basically, these two functions will form wacky shapes on your cartesean plane, you'll want to find intercepts, turning points and intersections. Looking for patterns as the values tend towards infinity.
That'd require some factoring and derivative-ing... trying to make things as simple as they can be (but not simpler than they have to be).
so you get f' and g'... your d/dx... as the case may be
you might then find some relationship/s you can exploit, 
usually a table of values can help (even if its an 'irregular' matrix).
there again, this might be more a question about Sigma Additivity. 
http://en.wikipedia.org/wiki/Sigma_additivity
that method might apply here, if all methods/approaches are open and on the table. (theres +infinity, -infinity involved)
It looks to be a finite bounded set, with a limited (albeit rather large) set of solutions.
http://en.wikipedia.org/wiki/Signed_measure
Arctan can be tricky when simplifying...
wiki Arctan
Good luck
