How to evaluate this limit?

Question

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 $$\lim _{x\to 0+}\left(\frac{\left[\ln\left(\frac{5+x^2}{5+4x}\right)\right]^6\ln\left(\frac{5+x^2}{1+4x}\right)}{\sqrt{5x^{10}+x^{11}}-\sqrt{5}x^5}\right)$$

Answer 1

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) $$ \ln 5\times \frac{4096}{15625}\times \frac{10}{\sqrt{5}}=\frac{8192 \ln (5)}{3125 \sqrt{5}}\ . $$

Answer 2

Let's try the elementary way. We have \begin{align} L &= \lim _{x \to 0^{+}}\left(\dfrac{\left[\log\left(\dfrac{5 + x^{2}}{5 + 4x}\right)\right]^{6} \log\left(\dfrac{5 + x^{2}}{1 + 4x}\right)}{\sqrt{5x^{10} + x^{11}} - \sqrt{5}x^5}\right)\notag\\ &= \lim _{x \to 0^{+}}\left(\dfrac{\left[\log\left(1 + \dfrac{x^{2} - 4x}{5 + 4x}\right)\right]^{6} \cdot\log 5}{\sqrt{5x^{10} + x^{11}} - \sqrt{5}x^5}\right)\notag\\ &= \log 5\lim _{x \to 0^{+}}\left(\dfrac{\left[\dfrac{\log\left(1 + \dfrac{x^{2} - 4x}{5 + 4x}\right)}{\dfrac{x^{2} - 4x}{5 + 4x}}\right]^{6}\left(\dfrac{x^{2} - 4x}{5 + 4x}\right)^{6}}{\sqrt{5x^{10} + x^{11}} - \sqrt{5}x^5}\right)\notag\\ &= \log 5\lim _{x \to 0^{+}}\left(\dfrac{1\cdot\left(\dfrac{x^{2} - 4x}{5 + 4x}\right)^{6}}{\sqrt{5x^{10} + x^{11}} - \sqrt{5}x^5}\right)\notag\\ &= \log 5\lim _{x \to 0^{+}}\left(\dfrac{x^{6}\left(\dfrac{x - 4}{5 + 4x}\right)^{6}}{x^{5}\{\sqrt{5 + x} - \sqrt{5}\}}\right)\notag\\ &= \log 5\lim _{x \to 0^{+}}\left(\dfrac{x - 4}{5 + 4x}\right)^{6}\cdot\frac{x}{\sqrt{5 + x} - \sqrt{5}}\notag\\ &= \left(\frac{4}{5}\right)^{6}\log 5\lim _{x \to 0^{+}}\frac{x(\sqrt{5 + x} + \sqrt{5})}{5 + x - 5}\notag\\ &= \left(\frac{4}{5}\right)^{6}2\sqrt{5}\log 5\notag\\ &= \frac{8192\sqrt{5}\log 5}{15625}\notag \end{align}