How to evaluate this limit? I have a problem with this limit, I have no idea how to compute it. Can you explain the method and the steps used(without L'Hopital if is possible)? Thanks
$$\lim _{x\to 0+}\left(\frac{\left[\ln\left(\frac{5+x^2}{5+4x}\right)\right]^6\ln\left(\frac{5+x^2}{1+4x}\right)}{\sqrt{5x^{10}+x^{11}}-\sqrt{5}x^5}\right)$$

Expand the denominator
$$
\sqrt{5}x^5 (\sqrt{1+x/5}-1)\sim \sqrt{5}x^5\left(\frac{x}{10}-\frac{x^2}{200}+\ldots\right)
$$
and the first term of the numerator
$$
\ln (5+x^2)=\ln5+\ln(1+x^2/5)=\ln5+x^2/5-x^4/50+\ldots
$$
$$
\ln(5+4x)=\ln 5+\log(1+4x/5)\sim \ln 5+4x/5-8x^2/25+\ldots
$$
Therefore
$$
\ln \left(\frac{5+x^2}{5+4x}\right)\sim -4x/5+(13/25)x^2+...\ ,
$$
which, raised to the power $6$, goes as $(4096/15625)x^6$. So, noting that $\ln((5+x^2)/(1+4x))$ has a finite limit $\ln 5$, the final value of the limit is (erasing the leading power $x^6$ upstairs and downstairs)
$$
\ln 5\times \frac{4096}{15625}\times \frac{10}{\sqrt{5}}=\frac{8192 \ln (5)}{3125 \sqrt{5}}\ .
$$
