1. ## Evaluate the limit

Hi
Wolfram Alpha has choked on the following:

$\lim_{x\to0}\frac{\ln\cos(x^2)+\sqrt[6]{1+3x^4}-1}{(\sqrt[]{\cos{x}}-\sqrt[4]{e^{-x^2}})\sin{(x^2)}\tan(e^{x^2}-1)}$

I feel powerless over the problem too. Any hints?

2. 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 $\infty$ 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 $\cos(x)$ behaves as 1 near $x = 0$, we get that $\ln(\cos(x))$ behaves like $\ln(1) = 0$ around $x =0$. Similarly $\sin(x^2)$ behaves like $x^2$ and $\tan(e^{x^2}-1)$ behaves like $e^{x^2}-1$. Also, $\sqrt{\cos(x)}$ behaves like $\sqrt{1} = 1$ around $x = 0$. If you use these you can see at least justify to yourself that the limit diverges.