Hi
Wolfram Alpha has choked on the following:
$\displaystyle \lim_{x\to0}\frac{\ln\cos(x^2)+\sqrt[6]{1+3x^4}-1}{(\sqrt[]{\cos{x}}-\sqrt[4]{e^{-x^2}})\sin{(x^2)}\tan(e^{x^2}-1)}$
I feel powerless over the problem too. Any hints?
That's a lot of input for that online form. I'm having trouble with that also. However, on my copy of Mathematica, it evaluates the limit as $\displaystyle \infty$ in like half a second.
The way you'd go about proving this by hand would be to use L'Hopital's rule repeatedly, although judging by how contrived the example is, it might require a LOT of work. Maybe there is a smart way, I dunno.
You can also see that it diverges if you consider the order of every function. Since $\displaystyle \cos(x)$ behaves as 1 near $\displaystyle x = 0$, we get that $\displaystyle \ln(\cos(x))$ behaves like $\displaystyle \ln(1) = 0$ around $\displaystyle x =0$. Similarly $\displaystyle \sin(x^2)$ behaves like $\displaystyle x^2$ and $\displaystyle \tan(e^{x^2}-1)$ behaves like $\displaystyle e^{x^2}-1$. Also, $\displaystyle \sqrt{\cos(x)}$ behaves like $\displaystyle \sqrt{1} = 1$ around $\displaystyle x = 0$. If you use these you can see at least justify to yourself that the limit diverges.