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

Using asymptotics for $\ x\to0$ we have:
$$\ \sqrt{1+x^2}≈1+\frac{x^2}{2}$$
$$\ \ln(1+x)≈x$$
$$\ \cos(x)≈1-\frac{x^2}{2}$$
So your limit gets:
$$\ \lim_{x\to0^+}\frac{\ln(x+\sqrt{1+x^2})}{\ln(\cos x)}≈\lim_{x\to0^+}\frac{\ln(1+x+\frac{x^2}{2})}{\ln(1-\frac{x^2}{2})}≈$$
$$≈\lim_{x\to0^+}\frac{x+\frac{x^2}{2}}{-\frac{x^2}{2}}=-\infty$$
