Another combined limit I've tried to get rid of those logarithms, but still, no result has came.
$$\lim_{x\to 0 x \gt 0} \frac{\ln(x+ \sqrt{x^2+1})}{\ln{(\cos{x})}}$$
Please help

Outline: Our expression, for $x\ne 0$, is equal to
$$\frac{\ln(x+\sqrt{x^2+1})}{x} \cdot\frac{x}{\ln(\cos x)}.$$ 
Note that 
$$\lim_{x\to 0} \frac{\ln(x+\sqrt{x^2+1})-0}{x}$$
is by definition the derivative of $\ln(x+\sqrt{x^2+1})$ at $x=0$. It is easy to compute that derivative. The only important thing is that it is positive.
Now we examine what happens to $\frac{x}{\ln(cos x)}$ as $x\to 0$. It is easier to look at what happens to $\frac{\ln(cosx)}{x}$. Again, think derivative.
